11footnotetext:  Department of Mathematics, Osaka City University, 3-3-138
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan.
<EMAIL_ADDRESS> Department of Mathematics,
Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan.
<EMAIL_ADDRESS>
# Asymptotic behavior of least energy solutions to the Finsler Lane-Emden
problem with large exponents

Sadaf Habibi1 and  Futoshi Takahashi2

###### Abstract.

In this paper we are concerned with the least energy solutions to the Lane-
Emden problem driven by an anisotropic operator, so-called the Finsler
$N$-Laplacian, on a bounded domain in $\mathbb{R}^{N}$. We prove several
asymptotic formulae as the nonlinear exponent gets large.

Key words: Finsler Lane-Emden problem, Finsler Laplacian, Least energy
solution
2010 Mathematics Subject Classification: 35J35, 35J60

##  1\. Introduction

Let $N\geq 2$ be an integer. In this paper, we study the following Lane-Emden
problem driven by an anisotropic operator $Q_{N}$:

(1.1) $\begin{cases}&-Q_{N}u=u^{p}\quad\mbox{in}\;\Omega,\\\
&u>0\quad\mbox{in}\;\Omega,\\\ &u=0\quad\mbox{on}\;\partial\Omega,\end{cases}$

where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $p>1$ is any
positive number, and $Q_{N}$ is a quasilinear operator, so-called the Finsler
$N$-Laplacian, defined by

$Q_{N}u=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}\left(H(\nabla
u)^{N-1}H_{\xi_{i}}(\nabla u)\right).$

Here $H\in C^{2}(\mathbb{R}^{N}\setminus\\{0\\})$ is any norm on
$\mathbb{R}^{N}$ and $H_{\xi_{i}}(\xi)=\frac{\partial
H(\xi)}{\partial\xi_{i}}$. We assume that $H^{N}\in C^{1}(\mathbb{R}^{N})$ and
${\rm Hess}\left(H^{N}(\xi)\right)$ is positive definite for any
$\xi\in\mathbb{R}^{N}$, $\xi\neq 0$. Note that $Q_{N}u$ can be written as

$Q_{N}u={\rm div}\left(\nabla_{\xi}(\frac{1}{N}H(\xi)^{N})\Big{|}_{\xi=\nabla
u}\right)=\sum_{i,j=1}^{N}a_{ij}(\nabla u)\frac{\partial^{2}u}{\partial
x_{i}\partial x_{j}},$

where $a_{ij}(\nabla u)={\rm
Hess}(\frac{1}{N}H^{N}(\xi))_{i,j}\Big{|}_{\xi=\nabla u}$. If $H(\xi)=|\xi|$
(the Euclidean norm), then $Q_{N}u$ coincides with the $N$-Laplacian
$\Delta_{N}u={\rm div}(|\nabla u|^{N-2}\nabla u)$ of a function $u$. In this
case, the problem (1.1) was treated by Ren and Wei [21] [22] when $N=2$, and
in [23] for general $N\geq 2$. Ren and Wei [23] considered the least energy
solution $u_{p}$ of the following quasilinear problem

$\displaystyle\begin{cases}&-\Delta_{N}u=u^{p}\quad\mbox{in}\;\Omega,\\\
&u>0\quad\mbox{in}\;\Omega,\\\ &u=0\quad\mbox{on}\;\partial\Omega,\end{cases}$

where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$. They studied
the asymptotic behavior of $u_{p}$ as the nonlinear exponent $p\to\infty$, and
proved that the least energy solutions remain bounded in $L^{\infty}$-norm
regardless of $p$. When the dimension $N=2$, they showed that the least energy
solutions must develop one “peak” in the interior of
$\Omega\subset\mathbb{R}^{2}$, that is, the shape of graph of $u_{p}$ looks
like a single spike as $p\to\infty$. Moreover they showed that this peak point
must be a critical point of the Robin function of the domain. For other
generalizations of this problem to various situations, see for example, [28],
[29], [30], [24], [25].

Now, main aim of the paper is to extend the results of Ren and Wei [21], [22],
[23] to the anisotropic problem (1.1).

As in [21], [22], [23], we restrict our attention to the least energy
solutions to (1.1) constructed as follows:

Consider the constrained minimization problem:

(1.2) $C_{p}=\inf\\{\int_{\Omega}H(\nabla u)^{N}dx:u\in
W^{1,N}_{0}(\Omega),\int_{\Omega}|u|^{L^{p+1}(\Omega)}dx=1\\}.$

Since the Sobolev imbedding $W^{1,N}_{0}(\Omega)\hookrightarrow
L^{p+1}(\Omega)$ is compact for any $p>1$, we have at least one minimizer
$\underline{u}_{p}$ for the problem (1.2), where $\underline{u}_{p}\in
W^{1,N}_{0}(\Omega),\|\underline{u}_{p}\|_{p+1}=1$. As $|\underline{u}_{p}|\in
W^{1,N}_{0}(\Omega)$ also achieves $C_{p}$, we may assume
$\underline{u}_{p}>0$. Note that $Q_{N}(cu)=c^{N-1}Q_{N}(u)$ for a constant
$c>0$. Thus if we define

$u_{p}=C_{p}^{\frac{1}{p+1-N}}\underline{u}_{p},$

then $u_{p}$ solves $\eqref{E_N}$ and $C_{p}=\int_{\Omega}H(\nabla
u_{p})^{N}dx/(\int_{\Omega}|u_{p}|^{p+1}dx)^{\frac{N}{p+1}}$. Standard
regularity argument implies that any weak solution $u\in W^{1,N}_{0}(\Omega)$
satisfies $u\in C^{1,\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$. We
call $u_{p}$ the least energy solution to (1.1).

Our first result is the following $L^{\infty}$-bound of least energy
solutions.

######  Theorem 1.1.

Let $u_{p}$ be a least energy solution to (1.1). Then there exist
$C_{1},C_{2}$ (independent of $p$), such that

$0<C_{1}\leq\|u_{p}\|_{L^{\infty}(\Omega)}\leq C_{2}<\infty$

for $p$ large enough.

Furthermore, we have

$\displaystyle\lim_{p\to\infty}p^{N-1}\int_{\Omega}H(\nabla
u_{p})^{N}dx=\lim_{p\to\infty}p^{N-1}\int_{\Omega}u_{p}^{p+1}dx=\left(\frac{Ne\beta_{N}}{N-1}\right)^{N-1}$

where $\beta_{N}=N(N\kappa_{N})^{\frac{1}{N-1}}$, $\kappa_{N}=|\mathcal{W}|$
is the volume (with respect to the $N$-dimensional Hausdorff measure) of the
unit Wulff ball associated with the dual norm $H^{0}$ of $H$:

$\mathcal{W}=\\{x\in\mathbb{R}^{N}:H^{0}(x)<1\\}.$

On the asymptotic behavior of the $L^{\infty}$-norm of $u_{p}$, we have

######  Theorem 1.2.

Let $u_{p}$ be a least energy solution to (1.1). Then it holds that

$1\leq\limsup_{p\to\infty}\|u_{p}\|_{L^{\infty}(\Omega)}\leq
e^{\frac{N-1}{N}}.$

To state further results, we need some definitions. Set

(1.3) $v_{p}=\frac{u_{p}}{(\int_{\Omega}u_{p}^{p}dx)^{\frac{1}{N-1}}}.$

Then $v_{p}$ is a weak solution of

(1.4)
$\begin{cases}-Q_{N}v_{p}=f_{p}(x)=\frac{u_{p}^{p}}{\int_{\Omega}u_{p}^{p}dx}\quad\text{in}\;\Omega,\\\
v_{p}>0\quad\text{in}\;\Omega,\\\
v_{p}=0\quad\text{on}\;\partial\Omega.\end{cases}$

For $X,Y\in\mathbb{R}^{N},X,Y\neq 0,X\neq Y$, we put

$d(X,Y)=\frac{\left(H^{N-1}(X)(\nabla_{\xi}H)(X)-H^{N-1}(Y)(\nabla_{\xi}H)(Y)\right)\cdot(X-Y)}{H^{N}(X-Y)},$

and

(1.5) $d_{N}=\inf\\{d(X,Y)\ |\ X,Y\in\mathbb{R}^{N},X,Y\neq 0,X\neq Y\\},$

where $X\cdot Y=\sum_{j=1}^{N}X_{j}Y_{j}$ denotes the usual inner product for
$X,Y\in\mathbb{R}^{N}$. As in [32] Lemma 5.1, we can obtain the estimate

$\min\\{\frac{\lambda}{\beta^{N}},1\\}\leq d_{N}\leq 1$

where $\lambda$ is the least eigenvalue of ${\rm
Hess}\left(\frac{1}{N}H^{N}(\xi)\right)$, which is positive by the assumption
(2.2) and $\beta$ is as in (2.3), see §2. Also define

(1.6) $\displaystyle
L_{0}=\limsup_{p\to\infty}\frac{p\left(\int_{\Omega}u_{p}^{p}dx\right)^{\frac{1}{N-1}}}{\left(\frac{N}{N-1}e^{\frac{N-1}{N}}\right)},\quad
L_{1}=d_{N}^{-\left(\frac{1}{N-1}\right)}L_{0}.$

For a sequence $v_{p_{n}}$ of $v_{p}$, we define the blow-up set $S$ of
$\\{v_{p_{n}}\\}$ as usual:

$S=\\{x\in\overline{\Omega}:\exists\mbox{a
subsequence}\;v_{p_{n}^{{}^{\prime}}},\exists\\{x_{n}\\}\subset\Omega\;\mbox{s.t.}\
x_{n}\to x\;\mbox{and}\;v_{p_{n}^{{}^{\prime}}}(x_{n})\to\infty\\}.$

In the following, $\sharp A$ denotes the cardinality of a set $A$ and
$[\cdot]$ denotes the Gauss symbol.

######  Theorem 1.3.

Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain. Then for any
sequence $v_{p_{n}}$ of $v_{p}$ with $p_{n}\to\infty$, the blow-up set $S$ of
$v_{p_{n}}$ is non-empty. Also there exists a subsequence (still denoted by
$v_{p_{n}}$) such that the estimate

$\sharp(S\cap\Omega)\leq\left[\frac{e^{\frac{N-1}{N}}}{d_{N}}\right]$

holds true for this subsequence.

Assume $S\cap\Omega=\\{x_{1},\cdots,x_{k}\\}\subset\Omega$. Then we have

  1. (i)

$f_{n}=\frac{u_{p_{n}}^{p_{n}}}{\int_{\Omega}u_{p_{n}}^{p_{n}}dx}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\sum_{i=1}^{k}\gamma_{i}\delta_{x_{i}}$

in the sense of Radon measures of $\Omega$, where

$\gamma_{i}\geq\left(\frac{\beta_{N}}{L_{1}}\right)^{N-1}$

and $\sum_{i=1}^{k}\gamma_{i}\leq 1$.

  2. (ii)

$v_{p_{n}}\to G$ in $C^{1}_{loc}(\Omega\setminus(S\cap\Omega))$ for some
function $G$ satisfying

$\begin{cases}-Q_{N}G=0\quad&\text{in}\,\Omega\setminus(S\cap\Omega),\\\
G=+\infty\quad&\text{on}\,S\cap\Omega,\\\
G=0\quad&\text{on}\,\partial\Omega\setminus(\partial\Omega\cap S).\end{cases}$

  3. (iii)

$\|u_{p_{n}}\|_{L^{\infty}(K)}\to 0$ as $n\to\infty$ for any compact set
$K\subset\Omega\setminus(S\cap\Omega)$.

In [23], Ren and Wei obtained an estimate of the number of interior blow-up
set

$\sharp(S\cap\Omega)\leq\left[\frac{1}{d_{N}}\left(\frac{N}{N-1}\right)^{N-1}\right]$

when $H(\xi)=|\xi|$ case. Since $e^{x}<\frac{1}{1-x}$ for $x\in(0,1)$, we
check that $e^{\frac{N-1}{N}}<\left(\frac{N}{N-1}\right)^{N-1}$ for all $N\geq
2$. Thus the estimate in Theorem 1.3 is better than that in [23] even when
$H(\xi)$ coincides with the Euclidean norm $|\xi|$. Also, Theorem 1.2 seems
new even for $H(\xi)=|\xi|$ and $N>2$ case.

Finally, we prove that if the blow-up set consists of one point, it must be an
interior point of $\Omega$.

######  Theorem 1.4.

Assume $\sharp S=1$ and $S=\\{x_{0}\\}$, $x_{0}\in\overline{\Omega}$. Then
$x_{0}\in\Omega$ must hold.

The organization of the paper is as follows: In §2, we recall basic properties
of the Finsler norm and collect useful lemmas about the Finsler $N$-Laplacian.
In §3, we obtain asymptotic formula for $C_{p}$ as $p\to\infty$, and prove the
latter half part of Theorem 1.1. In §4, we prove the $L^{\infty}$-bound of
least energy solutions in Theorem 1.1. In §5, we prove Theorem 1.2 using an
argument by Adimurthi and Grossi [1]. In §6, we prove Theorem 1.3. We use a
notion of $(L,\delta)$-regular, or irregular points, which was originally
introduced by Brezis and Merle [6]. Finally in §7, we prove Theorem 1.4 by
using a local Pohozaev identity and an idea by Santra and Wei [25].

##  2\. Notations and basic properties

Let $H$ be any norm on $\mathbb{R}^{N}$, i.e., $H$ is convex, $H(\xi)\geq 0$
and $H(\xi)=0$ if and only if $\xi=0$, and $H$ satisfies

(2.1) $H(t\xi)=|t|H(\xi),\quad\forall\xi\in\mathbb{R}^{N},\,\forall
t\in\mathbb{R}.$

By (2.1), $H$ must be even: $H(-\xi)=H(\xi)$ for all $\xi\in\mathbb{R}^{N}$.
Throughout of the paper, we also assume that $H\in
C^{2}(\mathbb{R}^{N}\setminus\\{0\\})$, $H^{N}\in C^{1}(\mathbb{R}^{N})$, and

(2.2) ${\rm Hess}\left(H^{N}(\xi)\right)$ is positive definite for any
$\xi\in\mathbb{R}^{N}$, $\xi\neq 0$.

Since all norms on $\mathbb{R}^{N}$ are equivalent to each other, we see the
existence of positive constants $\alpha$ and $\beta$ such that

(2.3) $\alpha|\xi|\leq H(\xi)\leq\beta|\xi|,\quad\xi\in\mathbb{R}^{N}.$

The dual norm of $H$ is the function $H^{0}:\mathbb{R}^{N}\to\mathbb{R}$
defined by

$H^{0}(x)=\sup_{\xi\in\mathbb{R}^{N}\setminus\\{0\\}}\frac{\xi\cdot
x}{H(\xi)}.$

It is well-known that $H^{0}$ is also a norm on $\mathbb{R}^{N}$ and satisfies
the inequality

$\frac{1}{\beta}|x|\leq H^{0}(x)\leq\frac{1}{\alpha}|x|,\quad\forall
x\in\mathbb{R}^{N}.$

The set

$\mathcal{W}=\\{x\in\mathbb{R}^{N}\,:\,H^{0}(x)<1\\}$

is called the Wulff ball, or the $H^{0}$-unit ball, and we denote
$\kappa_{N}=\mathcal{H}^{N}(\mathcal{W})$, where $\mathcal{H}^{N}$ denotes the
$N$-dimensional Hausdorff measure on $\mathbb{R}^{N}$. We also denote
$\mathcal{W}_{r}=\\{x\in\mathbb{R}^{N}\,|\,H^{0}(x)<r\\}$ for any $r>0$.

For a domain $\Omega\subset\mathbb{R}^{N}$ and a Borel set
$E\subset\mathbb{R}^{N}$, the anisotropic $H$-perimeter of a set $E$ with
respect to $\Omega$ is defined as

$P_{H}(E,\Omega)=\sup\left\\{\int_{E\cap\Omega}{\rm div}\sigma dx\ :\
\sigma\in C_{0}^{\infty}(\Omega,\mathbb{R}^{N}),H^{0}(\sigma(x))\leq
1\right\\}.$

If $E$ is Lipschitz, then it holds
$P_{H}(E,\Omega)=\int_{\Omega\cap\partial^{*}E}H(\nu)d\mathcal{H}^{N-1}$,
where $\partial^{*}E$ denotes the reduced boundary of the set $E$ and $\nu(x)$
is the measure theoretic outer unit normal of $\partial^{*}E$ (see [16]). Also
we have $P_{H}(\mathcal{W},\mathbb{R}^{N})=N\kappa_{N}$. For more explanation
about the anisotropic perimeter, see [3] and [5].

Here we just recall some properties of $H$ and $H^{0}$. These will be proven
by using the homogeneity property of $H$ and $H^{0}$, see [4] Lemma 2.1, and
Lemma 2.2.

######  Proposition 2.1.

Let $H$ be a Finsler norm on $\mathbb{R}^{N}$. Then the following properties
hold true:

  1. (1)

$|\nabla_{\xi}H(\xi)|\leq C$ for any $\xi\neq 0$.

  2. (2)

$\nabla_{\xi}H(\xi)\cdot\xi=H(\xi)$, $\nabla_{x}H(x)\cdot x=H(x)$ for any
$\xi\neq 0$, $x\neq 0$.

  3. (3)

$\left(\nabla_{\xi}H\right)(t\xi)=\frac{t}{|t|}\left(\nabla_{\xi}H\right)(\xi)$
for any $\xi\neq 0$, $t\neq 0$.

  4. (4)

$H\left(\nabla H^{0}(x)\right)=1$. $H^{0}\left(\nabla_{\xi}H(\xi)\right)=1$.

  5. (5)

$H^{0}(x)\left(\nabla_{\xi}H\right)\left(\nabla_{x}H^{0}(x)\right)=x$.

Finally, given a smooth function $u$ on $\mathbb{R}^{N}$, the Finsler Laplace
operator of $u$ (associated with $H$) is defined by

$\displaystyle Qu(x)$ $\displaystyle={\rm div}\left(H(\nabla
u(x))\left(\nabla_{\xi}H\right)(\nabla u(x))\right)$
$\displaystyle=\sum_{j=1}^{N}\frac{\partial}{\partial
x_{j}}\left(H(\xi)H_{\xi_{j}}(\xi)\Big{|}_{\xi=\nabla u(x)}\right)$

and, more generally, for any $1<q<\infty$, the Finsler $q$-Laplace operator
$Q_{q}$ by

$Q_{q}u(x)=\mbox{div}\left(H^{q-1}(\nabla u(x))(\nabla_{\xi}H)(\nabla
u(x))\right).$

If we assume that ${\rm Hess}(H^{q}(\xi))$ is positive definite on
$\mathbb{R}^{N}\setminus\\{0\\}$, $Q_{q}$ becomes a uniformly elliptic
operator locally. The Finsler $q$-Laplacian has been widely investigated in
literature by many authors in different settings, see [2], [5], [8], [9],
[10], [12], [13], [14], [17], [20] [34] and the references therein.

We collect here several useful facts.

######  Theorem 2.2.

(Finsler Trudinger-Moser inequality [32]) Let $\Omega$ be a bounded domain in
$\mathbb{R}^{N}$, $N\geq 2$. Let $u\in W^{1,N}_{0}(\Omega)$ satisfy
$\int_{\Omega}H(\nabla u)^{N}dx\leq 1$. Then there exists a constant $C$
depending only on the dimension $N$ such that

$\int_{\Omega}\exp\left(\beta|u|^{\frac{N}{N-1}}\right)dx\leq C|\Omega|$

holds for any $\beta\leq\beta_{N}=N(N\kappa_{N})^{\frac{1}{N-1}}$.
Furthermore, $\beta_{N}$ is optimal in the sense that there exists a sequence
$\\{u_{n}\\}\subset W^{1,N}_{0}(\Omega)$ with $\int_{\Omega}H(\nabla
u_{n})^{N}dx\leq 1$, such that
$\int_{\Omega}\exp\left(\beta|u_{n}|^{\frac{N}{N-1}}\right)dx\to+\infty$ as
$n\to\infty$ for $\beta>\beta_{N}$.

Next is the unique existence of the Green function for the Finsler
$p$-Laplacian.

######  Theorem 2.3.

([32]) Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain containing the
origin. Define $\Omega^{*}=\Omega\setminus\\{0\\}$ and

$\Gamma(x)=\begin{cases}&C(p,N)(H^{0}(x))^{\frac{p-N}{p-1}}\quad\text{for}\quad
1<p<N,\\\ &C(N)\log\frac{1}{H^{0}(x)}\quad\text{for}\quad p=N,\end{cases}$

where $C(p,N)=\frac{p-N}{p-1}(N\kappa_{N})^{-\frac{1}{p-1}}$ and
$C(N)=(N\kappa_{N})^{-\frac{1}{N-1}}$. Then there exists a unique function
$G(\cdot,0)\in C^{1,\alpha}(\Omega^{*})$ with $|\nabla G|\in L^{p-1}(\Omega)$,
$G/\Gamma\in L^{\infty}(\Omega)$, satisfying

$\displaystyle\begin{cases}-Q_{p}G(\cdot,0)=\delta_{0}&\quad\text{in}\
\Omega,\\\ G(\cdot,0)=0&\quad\text{on}\ \partial\Omega.\end{cases}$

Moreover, $g=G-\Gamma$ satisfies $g\in C(\Omega)$ and $\lim_{x\to
0}H^{0}(x)\nabla g(x)=0$.

We recall here useful regularity estimates which are valid for the Finsler
$N$-Laplacian equations, under the assumption (2.2); see Serrin [26],
Tolksdorf [31], DiBenedetto [15] and Lieberman [19].

######  Theorem 2.4.

Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain. Then the
following statements are true.

  1. (1)

Let $u\in W^{1,N}(\Omega)$ be a weak solution of $-Q_{N}u=f$ in $\Omega$,
where $f\in L^{q}(\Omega)$ for some $q>1$. Then for any subdomain
$\Omega^{\prime}\subset\subset\Omega$, there exists a constant
$C=C(\Omega,\Omega^{\prime},q,N)>0$ such that

$\|u\|_{L^{\infty}(\Omega^{\prime})}\leq
C\left(\|f\|_{L^{q}(\Omega)}+\|u\|_{L^{N}(\Omega)}\right)$

holds.

  2. (2)

Let $u\in W^{1,N}(\Omega)$ be a weak solution of $-Q_{N}u=f$ in $\Omega$.
Suppose $\|u\|_{L^{\infty}(\Omega)}\leq a$ and $\|f\|_{L^{\infty}(\Omega)}\leq
b$ for some $a,b<\infty$. Then $u\in C^{1,\alpha}_{loc}(\Omega)$ for some
$\alpha\in(0,1)$ and for any subdomain $\Omega^{\prime}\subset\subset\Omega$,
there exists a constant $C=C(\Omega,\Omega^{\prime},a,b,\alpha)>0$ such that

$\|u\|_{C^{1,\alpha}(\Omega^{\prime})}\leq C$

holds. If, in addition, $u$ satisfies the Dirichlet boundary condition
$u=\phi$ on $\partial\Omega$ where $\phi\in C^{1,\beta}(\partial\Omega)$,
$\beta\in(0,1)$, then $u\in C^{1,\alpha}(\overline{\Omega})$ for some
$\alpha\in(0,1)$ holds.

  3. (3)

(Harnack inequality) Let $u\in W^{1,N}(\Omega)$ be a nonnegative weak solution
of $-Q_{N}u=f$ in $\Omega$. Suppose $\|f\|_{L^{q}(\Omega)}\leq b$ for some
$q>1$. Then for any subdomain $\Omega^{\prime}\subset\subset\Omega$, there
exists a constant $C=C(\Omega,\Omega^{\prime},q,b)>0$ such that

$\sup_{x\in\Omega^{\prime}}u(x)\leq
C\left(1+\inf_{x\in\Omega^{\prime}}u(x)\right)$

holds.

Next is the result from [33] (Theorem 1.1 and Theorem 1.2).

######  Theorem 2.5.

(Finsler Brezis-Merle type inequality [33]) Let $\Omega$ be a bounded domain
in $\mathbb{R}^{N}$, $N\geq 2$.

  1. (1)

Suppose $u$ is a weak solution to

$\displaystyle\begin{cases}-Q_{N}u=f(x)&\quad\text{in}\ \Omega,\\\
u=0&\quad\text{on}\ \partial\Omega,\end{cases}$

where $f\in L^{1}(\Omega)$. Then for any $\varepsilon\in(0,\beta_{N})$ where
$\beta_{N}=N(N\kappa_{N})^{\frac{1}{N-1}}$, it holds that

$\int_{\Omega}\exp\left(\frac{(\beta_{N}-\varepsilon)|u(x)|}{\|f\|_{L^{1}(\Omega)}^{\frac{1}{N-1}}}\right)dx\leq\frac{\beta_{N}}{\varepsilon}|\Omega|.$

  2. (2)

Suppose $u$ and $v$ are weak solutions to

$-Q_{N}u=f(x)>0\quad\text{in}\ \Omega$

and

$-Q_{N}v=0\quad\text{in}\ \Omega\quad v=u\quad\text{on}\ \partial\Omega,$

respectively. Then for any $\varepsilon\in(0,\beta_{N})$, we have

$\int_{\Omega}\exp\left(\frac{(\beta_{N}-\varepsilon)d_{N}^{\frac{1}{N-1}}|u(x)-v(x)|}{\|f\|_{L^{1}(\Omega)}^{\frac{1}{N-1}}}\right)dx\leq\frac{|\Omega|}{\varepsilon},$

where $d_{N}$ is defined in (1.5).

Next is the Pohozaev identity for the Finsler $q$-Laplacian problem without
the boundary condition. This is a special case of much more general identity
proved in [11]. The identity below is known to hold for solutions in
$C^{1}(\overline{\Omega})\cap C^{2}(\Omega)$. The important point is that we
can remove the condition $u\in C^{2}(\Omega)$ with the cost of the convexity
and the $C^{1}(\mathbb{R}^{N})$-regularity of the map
$\mathbb{R}^{N}\ni\xi\mapsto H^{q}(\xi)$. This improvement is crucial for the
application to the Finsler Laplacian problem, since the best possible
regularity result of solutions is $C^{1,\alpha}$, not $C^{2}$, see Theorem
2.4.

######  Theorem 2.6.

([11]) Let $1<q<\infty$. Let $u\in C^{1}(\overline{\Omega})$ be a weak
solution of $-Q_{q}u=f(u)$ in $\Omega$, where $\Omega\subset\mathbb{R}^{N}$ is
a domain with the boundary of class $C^{1}$, and $f\in
C(\mathbb{R},\mathbb{R})$. Assume the map $\mathbb{R}^{N}\ni\xi\mapsto
H^{q}(\xi)$ is convex and belongs to $C^{1}(\mathbb{R}^{N})$. Then the
identity

$\displaystyle N\int_{\Omega}F(u)dx$
$\displaystyle-\left(\frac{N-q}{q}\right)\int_{\Omega}H^{q}(\nabla u)dx$
$\displaystyle=\int_{\partial\Omega}F(u)(x-y)\cdot\nu(x)ds_{x}$
$\displaystyle-\frac{1}{q}\int_{\partial\Omega}H^{q}(\nabla
u)(x-y)\cdot\nu(x)ds_{x}$
$\displaystyle+\int_{\partial\Omega}\left(H^{q-1}(\nabla
u)(\nabla_{\xi}H)(\nabla u)\cdot\nu(x)\right)((x-y)\cdot\nu(x))ds_{x}$

holds true for any $y\in\mathbb{R}^{N}$. Here $\nu$ is the outer unit normal
of $\partial\Omega$ and $F(s)=\int_{0}^{s}f(t)dt$.

###### Proof.

Indeed, since $\mathcal{L}(x,s,\xi)=\frac{1}{q}H^{q}(\xi)-F(s)$ is of the
“splitting” form, $F\in C^{1}(\mathbb{R})$, and $\xi\mapsto H^{q}(\xi)$ is
convex and in $C^{1}(\mathbb{R}^{N})$, Lemma 5, thus the equation (3) in [11]
holds as it is. Also, if we do not impose the boundary condition $u=0$ on
$\partial\Omega$ (and put $f=0$ there) in Lemma 2 in [11], we obtain the
identity

$\displaystyle\int_{\partial\Omega}\mathcal{L}(x,u,\nabla
u)(h\cdot\nu)ds_{x}-\sum_{i,j=1}^{N}\int_{\partial\Omega}h_{j}D_{\xi_{i}}\mathcal{L}(x,u,\nabla
u)D_{x_{j}}u\nu_{i}ds_{x}$ $\displaystyle\hskip 20.00003pt=\int_{\Omega}({\rm
div}h)\mathcal{L}(x,u,\nabla
u)dx-\sum_{i,j=1}^{N}\int_{\partial\Omega}D_{i}h_{j}D_{\xi_{i}}\mathcal{L}(x,u,\nabla
u)D_{x_{j}}udx$

for every $h\in C^{1}(\overline{\Omega},\mathbb{R}^{N})$. Inserting $h(x)=x$
leads to the claim.

Finally, we prove the following simple lemma.

######  Lemma 2.7.

Let $u\in W^{1,N}_{0}(\Omega)$ be a weak solution to $-Q_{N}u=f(u)$ in
$\Omega\subset\mathbb{R}^{N}$, where $f:\mathbb{R}\to\mathbb{R}$ is
continuous. Let $a,c>0$, $d\in\mathbb{R}$ and $b\in\mathbb{R}^{N}$. Then
$v(x)=cu(ax+b)+d$, $x\in\Omega_{a,b}=\frac{\Omega-b}{a}$ is a weak solution to

$-Q_{N}v=a^{N}c^{N-1}f\left(\frac{v-d}{c}\right)\quad\text{in}\
\Omega_{a,b},\quad v=0\quad\text{on}\ \partial\Omega_{a,b}.$

###### Proof.

For $x\in\Omega_{a,b}$, put $y=ax+b\in\Omega$. Then for any $\phi\in
C_{0}^{\infty}(\Omega_{a,b})$, $\tilde{\phi}(y)=\phi(x)$ belongs to
$C_{0}^{\infty}(\Omega)$. Therefore we have

$\displaystyle\int_{\Omega_{a,b}}H^{N-1}(\nabla v(x))(\nabla_{\xi}H)(\nabla
v(x))\cdot\nabla\phi(x)dx$ $\displaystyle=\int_{\Omega_{a,b}}H^{N-1}(ca(\nabla
u)(ax+b))(\nabla_{\xi}H)(ca(\nabla u)(ax+b))\cdot\nabla\phi(x)dx$
$\displaystyle=\int_{\Omega}c^{N-1}a^{N-1}H^{N-1}(\nabla
u(y))(\nabla_{\xi}H)((\nabla u(y))\cdot a\nabla\tilde{\phi}(y)a^{-N}dy$
$\displaystyle=c^{N-1}\int_{\Omega}H^{N-1}(\nabla u(y))(\nabla_{\xi}H)(\nabla
u(y))\cdot\nabla\tilde{\phi}(y)dy$
$\displaystyle=c^{N-1}\int_{\Omega}f(u(y))\tilde{\phi}(y)dy$
$\displaystyle=c^{N-1}\int_{\Omega_{a,b}}f\left(\frac{v(x)-d}{c}\right)\phi(x)a^{N}dx,$

where we have used (2.1) and Proposition 2.1 (3). Thus we see

$\displaystyle\int_{\Omega_{a,b}}H^{N-1}(\nabla v(x))(\nabla_{\xi}H)(\nabla
v(x))\cdot\nabla\phi(x)dx=a^{N}c^{N-1}\int_{\Omega_{a,b}}f\left(\frac{v(x)-d}{c}\right)\phi(x)dx.$

This holds true for any $\phi\in C_{0}^{\infty}(\Omega_{a,b})$, which implies
Lemma.

##  3\. Asymptotic estimate for $C_{p}$

In this section, first by using the Finsler Trudinger-Moser inequality Theorem
2.2, we establish the refined Sobolev embedding.

######  Lemma 3.1.

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain. For any $t\geq 2$,
there exists $D_{t}>0$ such that for any $u\in W^{1,N}_{0}(\Omega)$,

$\|u\|_{L^{t}(\Omega)}\leq D_{t}t^{\frac{N-1}{N}}\|H(\nabla
u)\|_{L^{N}(\Omega)}$

holds true. Furthermore, we have

$\lim_{t\to\infty}D_{t}=\left(\frac{1}{N\kappa_{N}^{1/N}}\right)\left(\frac{N-1}{Ne}\right)^{\frac{N-1}{N}}.$

###### Proof.

Let $u\in W^{1,N}_{0}(\Omega)$. By the elementary inequality
$\frac{x^{s}}{\Gamma(s+1)}\leq e^{x}$ for $x\geq 0$ and $s\geq 0$, where
$\Gamma(s)$ is the Gamma function, and the Finsler Trudinger-Moser inequality,
we have

$\displaystyle\frac{1}{\Gamma(\frac{N-1}{N}t+1)}\int_{\Omega}|u|^{t}dx$
$\displaystyle=\frac{1}{\Gamma(\frac{N-1}{N}t+1)}\int_{\Omega}\left(\beta_{N}\left(\frac{|u|}{\|H(\nabla
u)\|_{L^{N}(\Omega)}}\right)^{\frac{N}{N-1}}\right)^{\frac{N-1}{N}t}dx\beta_{N}^{-\frac{N-1}{N}t}\|H(\nabla
u)\|_{L^{N}(\Omega)}^{t}$
$\displaystyle\leq\int_{\Omega}\exp\left(\left(\beta_{N}\frac{|u(x)|}{\|\nabla
u\|_{L^{N}(\Omega)}}\right)^{\frac{N}{N-1}}\right)dx\beta_{N}^{-\frac{N-1}{N}t}\|H(\nabla
u)\|_{L^{N}(\Omega)}^{t}$ $\displaystyle\leq
C|\Omega|\beta_{N}^{-\frac{N-1}{N}t}\|H(\nabla u)\|_{L^{N}(\Omega)}^{t}.$

Put

$D_{t}=\Gamma\left(\frac{N-1}{N}t+1\right)^{1/t}C^{1/t}|\Omega|^{1/t}\beta_{N}^{-\frac{N-1}{N}}t^{-\frac{N-1}{N}}.$

Then we have

$\|u\|_{L^{t}(\Omega)}\leq D_{t}t^{\frac{N-1}{N}}\|H(\nabla
u)\|_{L^{N}(\Omega)}.$

Stirling’s formula implies that

$\left(\Gamma\left(\frac{(N-1)t}{N}+1\right)\right)^{\frac{1}{t}}\sim\left(\frac{N-1}{Ne}\right)^{\frac{N-1}{N}}t^{\frac{N-1}{N}}$

as $t\to\infty$. So we have

$\lim_{t\to\infty}D_{t}=\beta_{N}^{-\frac{N-1}{N}}\left(\frac{N-1}{Ne}\right)^{\frac{N-1}{N}}=\left(\frac{1}{N\kappa_{N}^{1/N}}\right)\left(\frac{N-1}{Ne}\right)^{\frac{N-1}{N}},$

which is a desired result.

Recall that $C_{p}$ is defined in (1.2). Using the above Lemma and energy
comparison, we get the following.

######  Proposition 3.2.

We have

$\lim_{p\to\infty}p^{N-1}C_{p}=\left(\frac{Ne}{N-1}\beta_{N}\right)^{N-1}.$

where $\beta_{N}=N(N\kappa_{N})^{\frac{1}{N-1}}$.

###### Proof.

Lower bound
$\liminf_{p\to\infty}(p+1)^{N-1}C_{p}\geq\left(\frac{Ne}{N-1}\beta_{N}\right)^{N-1}$
is a direct consequence of Lemma 3.1 and the fact

(3.1) $C_{p}=\frac{\|H(\nabla
u_{p})\|_{L^{N}(\Omega)}^{N}}{\|u_{p}\|_{L^{p+1}(\Omega)}^{N}}$

for least energy solutions $u_{p}$.

Therefore we must prove only the upper bound. We will do this by constructing
a suitable test function for the value $C_{p}$.

We may assume that $0\in\Omega$ and $\mathcal{W}_{L}\subset\Omega$ where
$\mathcal{W}_{L}=\\{x\in\mathbb{R}^{N}:H^{0}(x)<L\\}$. For $0<l<L$, consider
the Finsler Moser function

$m_{l}(x)=\frac{1}{(N\kappa_{N})^{1/N}}\begin{cases}\left(\log\frac{L}{l}\right)^{\frac{N-1}{N}},\quad&0\leq
H^{0}(x)\leq l,\\\
\frac{\log\frac{L}{H^{0}(x)}}{(\log\frac{L}{l})^{\frac{1}{N}}},\quad&l\leq
H^{0}(x)\leq L,\\\ 0,\quad&L\leq H^{0}(x).\end{cases}$

We check that the Moser function $m_{l}\in W^{1,N}_{0}(\Omega)$ and
$\|H(\nabla m_{l})\|_{L^{N}(\Omega)}=1$. Also it is easily checked that

$\left(\int_{\Omega}m_{l}^{p+1}dx\right)^{\frac{1}{p+1}}\geq\left(\int_{\mathcal{W}_{l}}m_{l}^{p+1}dx\right)^{\frac{1}{p+1}}\geq\frac{1}{(N\kappa_{N})^{1/N}}\left(\log\frac{L}{l}\right)^{\frac{N-1}{N}}\left(l^{N}\kappa_{N}\right)^{\frac{1}{p+1}}.$

Choosing $l=L\exp\left(-(\frac{N-1}{N^{2}})(p+1)\right)$, we have

$\|m_{l}\|_{L^{p+1}(\Omega)}\geq\frac{1}{(N\kappa_{N})^{1/N}}\left(\frac{N-1}{N^{2}}\right)^{\frac{N-1}{N}}e^{-\frac{N-1}{N}}(p+1)^{\frac{N-1}{N}}\left(L^{N}\kappa_{N}\right)^{\frac{1}{p+1}}.$

and

$C_{p}\leq\frac{\|H(\nabla
m_{l})\|_{L^{N}(\Omega)}^{N}}{\|m_{l}\|_{L^{p+1}(\Omega)}^{N}}\leq
N\kappa_{N}\left(\frac{N^{2}e}{N-1}\right)^{N-1}(p+1)^{-(N-1)}(L^{N}\kappa_{N})^{-\frac{N}{p+1}},$

which implies
$\limsup_{p\to\infty}(p+1)^{N-1}C_{p}\leq\left(\frac{Ne}{N-1}\beta_{N}\right)^{N-1}$.

Since

$\int_{\Omega}H(\nabla u_{p})^{N}dx=\int_{\Omega}u_{p}^{p+1}dx$

and (3.1), we have the following lemma.

######  Lemma 3.3.

$\lim_{p\to\infty}p^{N-1}\int_{\Omega}H(\nabla
u_{p})^{N}dx=\lim_{p\to\infty}p^{N-1}\int_{\Omega}u_{p}^{p+1}dx=\left(\frac{Ne}{N-1}\beta_{N}\right)^{N-1}.$

##  4\. Proof of Theorem 1.1

To obtain a lower bound for $\|u_{p}\|_{L^{\infty}(\Omega)}$, define the first
eigenvalue of the Finsler $N$-Laplacian $Q_{N}$:

$\lambda_{1}(\Omega)=\inf\\{\int_{\Omega}H(\nabla u)^{N}dx:u\in
W^{1,N}_{0}(\Omega),\int_{\Omega}|u|^{N}dx=1\\}.$

It is known that $0<\lambda_{1}(\Omega)<\infty$ and

$\int_{\Omega}u_{p}^{p+1}dx=\int_{\Omega}H(\nabla
u_{p})^{N}dx\geq\lambda_{1}(\Omega)\int_{\Omega}u_{p}^{N}dx.$

Thus

$\int_{\Omega}(u_{p}^{p+1}-\lambda_{1}(\Omega)u_{p}^{N})dx\geq 0,$

which implies

(4.1) $\|u_{p}\|_{L^{\infty}(\Omega)}^{p+1-N}\geq\lambda_{1}(\Omega).$

To obtain a uniform upper bound of $\|u_{p}\|_{L^{\infty}(\Omega)}$, we use an
argument with the coarea formula and the Finsler isoperimetric inequality in
$\mathbb{R}^{N}$. Set

$\displaystyle\gamma_{p}=\max_{x\in\Omega}u_{p}(x),$
$\displaystyle\Omega_{t}=\\{x\in\Omega:u_{p}(x)>t\\},$
$\displaystyle\mathcal{A}=\\{x\in\Omega:u_{p}(x)>\frac{\gamma_{p}}{2}\\}.$

By Lemma 3.1 with $t=\frac{Np}{N-1}$ and by Lemma 3.3, we have

$\left(\int_{\Omega}u_{p}^{\frac{Np}{N-1}}dx\right)^{\frac{N-1}{Np}}\leq
D_{\frac{Np}{N-1}}\left(\frac{Np}{N-1}\right)^{\frac{N-1}{N}}\|H(\nabla
u_{p})\|_{L^{N}(\Omega)}\leq M$

where $M$ is independent of $p$ if $p$ large. From this and Chebyshev’s
inequality, we have

(4.2) $\left(\frac{\gamma_{p}}{2}\right)^{\frac{Np}{N-1}}|\mathcal{A}|\leq
M^{\frac{Np}{N-1}}.$

On the other hand, by approximating the constant $1$ on $\Omega_{t}$ by
$C_{0}^{\infty}$-functions, we have

$-\int_{\Omega_{t}}{\rm div}\left(H(\nabla u_{p})^{N-1}(\nabla_{\xi}H)(\nabla
u_{p})\right)dx=\int_{\Omega_{t}}u_{p}^{p}dx.$

Thus integration by parts leads to

$\displaystyle\int_{\Omega_{t}}u_{p}^{p}dx$
$\displaystyle=-\int_{\partial\Omega_{t}}H(\nabla
u_{p})^{N-1}(\nabla_{\xi}H)(\nabla u_{p})\cdot\nu ds$ (4.3)
$\displaystyle=\int_{\partial\Omega_{t}}\frac{H(\nabla
u_{p})^{N-1}(\nabla_{\xi}H)(\nabla u_{p})\cdot\nabla u_{p}}{|\nabla u_{p}|}ds$
$\displaystyle=\int_{\partial\Omega_{t}}\frac{H(\nabla u_{p})^{N}}{|\nabla
u_{p}|}ds,$

since the outer unit normal $\nu$ to $\partial\Omega_{t}$ is
$\nu=-\frac{\nabla u_{p}}{|\nabla u_{p}|}$. Here we used Proposition 2.1 (3)
in the last equality. Coarea formula implies

$|\Omega_{t}|=\int_{\Omega_{t}}1dx=\int_{t}^{\infty}\int_{\\{u_{p}=s\\}}\frac{ds}{|\nabla
u_{p}|}.$

Thus

(4.4) $-\frac{d}{dt}|\Omega_{t}|=\int_{\partial\Omega_{t}}\frac{ds}{|\nabla
u_{p}|}.$

By (4), (4.4), and the Schwartz inequality, we have

$\displaystyle\left(-\frac{d}{dt}|\Omega_{t}|\right)^{N-1}\int_{\Omega_{t}}u_{p}^{p}dx$
$\displaystyle=\left(\int_{\partial\Omega_{t}}\frac{1}{|\nabla
u_{p}|}ds\right)^{N-1}\left(\int_{\partial\Omega_{t}}\frac{H^{N}(\nabla
u_{p})}{|\nabla u_{p}|}ds\right)$ (4.5)
$\displaystyle\geq\left(\int_{\partial\Omega_{t}}\frac{H(\nabla
u_{p})}{|\nabla u_{p}|}ds\right)^{N}$
$\displaystyle=\left(\int_{\partial\Omega_{t}}H(\nu)ds\right)^{N}$
$\displaystyle=P_{H}(\Omega_{t},\mathbb{R}^{N})^{N}\geq
N^{N}\kappa_{N}|\Omega_{t}|^{N-1}.$

In the last inequality of (4), we used the Finsler isoperimetric inequality in
$\mathbb{R}^{N}$ [3], [27], [18]:

(4.6) $P_{H}(E,\mathbb{R}^{N})\geq
N\kappa_{N}^{\frac{1}{N}}|E|^{\frac{N-1}{N}}$

for any set of finite perimeter $E\subset\mathbb{R}^{N}$ with respect to $H$.

Now, define $r(t)>0$ such that

$|\Omega_{t}|=\kappa_{N}r^{N}(t).$

Then

$\frac{d}{dt}|\Omega_{t}|=N\kappa_{N}r^{N-1}(t)r^{\prime}(t).$

Note that $r^{\prime}(t)<0$. Putting this in (4), we have

$\displaystyle\left(-N\kappa_{N}r^{N-1}(t)\frac{dr}{dt}(t)\right)^{N-1}\int_{\Omega_{t}}u_{p}^{p}dx\geq
N^{N}\kappa_{N}|\Omega_{t}|^{N-1},$
$\displaystyle\left(-\frac{dr}{dt}\right)^{N-1}\int_{\Omega_{t}}u_{p}^{p}dx\geq(N\kappa_{N})r^{N-1},$
$\displaystyle-\frac{dt}{dr}\leq\left(\int_{\Omega_{t}}u_{p}^{p}dx\right)^{\frac{1}{N-1}}(N\kappa_{N})^{-\frac{1}{N-1}}r^{-1}$
$\displaystyle\leq
Cr^{-1}\gamma_{p}^{\frac{p}{N-1}}|\Omega_{t}|^{\frac{1}{N-1}}=C\gamma_{p}^{\frac{p}{N-1}}r^{\frac{1}{N-1}},$

where $C$ is a constant dependent only on $N$ and varies from line to line.
Integrating the last inequality from $r=0$ to $r=r_{0}$, we have

$t(0)-t(r_{0})\leq C\gamma_{p}^{\frac{p}{N-1}}r_{0}^{\frac{N}{N-1}}.$

Choose $r_{0}$ such that $t(r_{0})=\frac{\gamma_{p}}{2}$. Then the above
inequality implies

$\displaystyle\gamma_{p}\leq
C\gamma_{p}^{\frac{p}{N-1}}r_{0}^{\frac{N}{N-1}},\quad\text{i.e.,}\quad\gamma_{p}\leq
C\gamma_{p}^{\frac{p}{N-1}}|\mathcal{A}|^{\frac{1}{N-1}}.$

Combining this with (4.2), we have

$\displaystyle\gamma_{p}\leq
C\gamma_{p}^{\frac{p}{N-1}}\left(\frac{M^{\frac{Np}{N-1}}}{\left(\frac{\gamma_{p}}{2}\right)^{\frac{Np}{N-1}}}\right)^{\frac{1}{N-1}}=C\gamma_{p}^{-\frac{p}{(N-1)^{2}}}M^{\frac{Np}{(N-1)^{2}}},$
$\displaystyle\gamma_{p}^{1+\frac{p}{(N-1)^{2}}}\leq
CM^{\frac{Np}{(N-1)^{2}}},$ $\displaystyle\gamma_{p}\leq
C^{\frac{(N-1)^{2}}{(N-1)^{2}+p}}M^{\frac{Np}{(N-1)^{2}+p}}.$

From this, we conclude that there exists $C>0$ (independent of $p$) such that
$\gamma_{p}\leq C$ for $p$ large.

The latter half part of Theorem 1.1 is already proven in Lemma 3.3. Thus we
have completed the proof of Theorem 1.1.

From Theorem 1.1, we have the following consequence.

######  Corollary 4.1.

There exist $C,C^{\prime}>0$ independent of $p$ large such that

$C\leq p^{N-1}\int_{\Omega}u_{p}^{p}dx\leq C^{\prime}$

holds true.

###### Proof.

By Theorem 1.1, we have

$\frac{1}{C_{2}}p^{N-1}\int_{\Omega}u_{p}^{p+1}dx\leq\frac{\|u_{p}\|_{L^{\infty}(\Omega)}}{C_{2}}p^{N-1}\int_{\Omega}u_{p}^{p}dx\leq
p^{N-1}\int_{\Omega}u_{p}^{p}dx$

where $C_{2}$ is as in Theorem 1.1. The left-hand side of the above inequality
is bounded from below by a positive constant by Lemma 3.3. On the other hand,
Hölder’s inequality implies

$p^{N-1}\int_{\Omega}u_{p}^{p}dx\leq\left(p^{N-1}\int_{\Omega}u_{p}^{p+1}dx\right)^{\frac{p}{p+1}}p^{\frac{1}{p+1}}|\Omega|^{\frac{1}{p+1}}$

and the right-hand side of the above inequality is bounded from above by Lemma
3.3. This proves the conclusion.

##  5\. Proof of Theorem 1.2

In this section, we prove Theorem 1.2. Since
$\limsup_{p\to\infty}\|u_{p}\|_{L^{\infty}(\Omega)}\geq 1$ immediately follows
from (4.1) (this is true for any solution sequence, not necessary least energy
solutions), we just need to prove
$\limsup_{p\to\infty}\|u_{p}\|_{L^{\infty}(\Omega)}\leq e^{\frac{N-1}{N}}$.
For this purpose, we follow the argument by Adimurthi and Grossi [1].

Let $x_{p}\in\Omega$ be a point so that the least energy solution to (1.1)
takes its maximum: $u_{p}(x_{p})=\|u_{p}\|_{L^{\infty}(\Omega)}$. As in [1],
We make a change of variable

(5.1)
$z_{p}(x)=\frac{p}{u_{p}(x_{p})}\left(u_{p}(\varepsilon_{p}x+x_{p})-u_{p}(x_{p})\right),\quad
x\in\Omega_{p}=\frac{\Omega-x_{p}}{\varepsilon_{p}},$

where $\varepsilon_{p}>0$ is defined so that

(5.2) $\varepsilon_{p}^{N}p^{N-1}u_{p}(x_{p})^{p+1-N}\equiv 1.$

By Theorem 1.1, we see $\varepsilon_{p}\to 0$ as $p\to\infty$. Since $u_{p}$
is a weak solution to (1.1), $z_{p}$ is a weak solution to

(5.3)
$\displaystyle\begin{cases}&-Q_{N}z_{p}=\left(1+\frac{z_{p}}{p}\right)^{p}\quad\mbox{in}\;\Omega_{p},\\\
&z_{p}|_{\partial\Omega_{p}}=-p,\\\
&\max_{x\in\overline{\Omega}_{n}}z_{n}(x)=z_{n}(0)=0,\\\ &-p<z_{p}\leq
0\quad\mbox{in}\;\Omega_{p}\\\ \end{cases}$

by Lemma 2.7. We want to pass to the limit as $p\to\infty$ in (5.3). For this
purpose, take any ball $B_{R}(0)\subset\Omega_{p}$ centered at the origin and
radius $R$. Consider

$\displaystyle\begin{cases}&-Q_{N}w_{p}=\left(1+\frac{z_{p}}{p}\right)^{p}\quad\mbox{in}\;B_{R}(0),\\\
&w_{p}|_{\partial B_{R}(0)}=0.\end{cases}$

Comparison principle for $-Q_{N}$ (see for example, [33] Theorem 3.1) and
Serrin’s elliptic estimate Theorem 2.4 yield that $0\leq w_{p}\leq C$ on
$B_{R}(0)$ where $C$ is a constant independent of $p$. Set
$\psi_{p}(x)=w_{p}(x)-z_{p}(x),x\in B_{R}(0)$. Then $\psi_{p}$ is a
nonnegative in $B_{R}(0)$ and $\psi_{p}(0)=w_{p}(0)-z_{p}(0)=w_{p}(0)\leq C$
uniformly in $p$. Moreover, we have

$0=-(Q_{N}w_{p}-Q_{N}z_{p})=-\tilde{Q}_{N}(w_{p}-z_{p})=-\tilde{Q}_{N}\psi_{p}$

where

$\displaystyle\tilde{Q}_{N}(w_{p}-z_{p})$
$\displaystyle=\sum_{i,j=1}^{N}\frac{\partial}{\partial
x_{i}}\left[\int_{0}^{1}\frac{1}{N}\frac{\partial^{2}H^{N}}{\partial\xi_{i}\partial\xi_{j}}(t\nabla
w_{p}+(1-t)\nabla z_{p})dt\frac{\partial}{\partial
x_{j}}(w_{p}(x)-z_{p}(x))\right].$

Thanks to the assumption that ${\rm Hess}H^{N}(\xi)$ is positive definite,
$\tilde{Q}_{N}$ is a quasilinear elliptic differential operator. Thus we can
apply Serrin’s Harnack inequality (Theorem 2.4 (3)) to $\psi_{p}$, which
implies that there exists $C=C(R,r)>0$ for any $0<r<R$ such that

$\sup_{B_{r}(0)}\psi_{p}(x)\leq C\left(1+\inf_{x\in
B_{r}(0)}\psi_{p}(x)\right)\leq C(1+\psi_{p}(0))=C(1+w_{p}(0))\leq C.$

Thus we have

$0\geq z_{p}(x)=w_{p}(x)-\psi_{p}(x)\geq-C$

for $x\in B_{r}(0)$. Since $0<r<R$ is arbitrary, we have $\\{|z_{p}|\\}\subset
L^{\infty}_{loc}(B_{R}(0)))$ is uniformly bounded in $p$. Again Serrin’s
regularity estimate implies that $\\{z_{p}\\}$ is bounded in
$C^{1,\alpha}_{loc}(B_{R}(0)))$ for any $R>0$ uniformly in $p$.

Now, we consider two cases:

Case (i): $\frac{{\rm
dist}(x_{p},\partial\Omega_{p})}{\varepsilon_{p}}\to+\infty$

Case (ii): $\frac{{\rm dist}(x_{p},\partial\Omega_{p})}{\varepsilon_{p}}$ is
bounded and

$\Omega_{p}\to\mathbb{R}^{N}_{+}(s_{0})=\\{x=(x^{\prime},x_{N})\in\mathbb{R}^{N}\,:\,x_{N}>s_{0}\\}\quad(p\to\infty)$

for some $s_{0}$.

In the case (i), note that $\Omega_{p}\to\mathbb{R}^{N}$ as $p\to\infty$.
Hence by the Ascoli-Arzelá theorem, we know that (up to a subsequence),
$\\{z_{p}\\}$ converges to some function $z\in C^{1}(\mathbb{R}^{N})$ and $z$
satisfies

$-Q_{N}z=e^{z}\quad\text{in}\,\mathbb{R}^{N}.$

Now we claim that $\int_{\mathbb{R}^{N}}e^{z}dx<+\infty$. In fact, since
$z_{p}\to z$ in $C^{1}_{loc}(\mathbb{R}^{N})$, we obtain

$1_{\Omega_{p}}(x)\left(1+\frac{z_{p}(x)}{p}\right)^{p}\to e^{z(x)}$

pointwisely for $x\in\mathbb{R}^{N}$, where $1_{\Omega_{p}}$ is the
characteristic function of $\Omega_{p}$. By using Fatou’s lemma and Hölder’s
inequality, we deduce

$\displaystyle\int_{\mathbb{R}^{N}}e^{z}dx$
$\displaystyle\leq\liminf_{p\to\infty}\int_{\Omega_{p}}\left(1+\frac{z_{p}(x)}{p}\right)^{p}dx$
$\displaystyle\leq\lim_{p\to\infty}\frac{p^{N-1}}{(u_{p}(x_{p}))^{N-1}}\int_{\Omega}(u_{p}(y))^{p}dy$
$\displaystyle\leq\lim_{p\to\infty}\frac{p^{N-1}}{(u_{p}(x_{p}))^{N-1}}\left(\int_{\Omega}(u_{p}(y))^{p+1}dy\right)^{p/(p+1)}|\Omega|^{1/(p+1)}$
$\displaystyle\leq C<\infty$

where we have used the facts that
$\int_{\Omega}u_{p}^{p+1}dy=\frac{O(1)}{p^{N-1}}$ by Lemma 3.3 and
$u_{p}(x_{p})\geq C_{1}>0$ by Theorem 1.1. Hence, we check that the limit
function satisfies

(5.4) $\begin{cases}&-Q_{N}z=e^{z}\quad\text{in}\,\mathbb{R}^{N},\\\ &z\leq
0,\quad\text{in}\,\mathbb{R}^{N},\\\
&\int_{\mathbb{R}^{N}}e^{z}dx<\infty.\end{cases}$

In the case (ii), almost the same proof works, and we see that the limit
function $z$ is a solution of

(5.5)
$\begin{cases}&-Q_{N}z=e^{z}\quad\text{in}\,\mathbb{R}^{N}_{+}(s_{0}),\\\
&z\leq 0,\quad\text{in}\,\mathbb{R}^{N}_{+}(s_{0}),\\\
&z=-\infty,\quad\text{on}\,\partial\mathbb{R}^{N}_{+}(s_{0}),\\\
&\int_{\mathbb{R}^{N}_{+}(s_{0})}e^{z}dx<\infty.\end{cases}$

Now we prove the following lemma. The case $N=2$ was treated by Ding (see [7])
when $H(\xi)=|\xi|$, and by Wang and Xia [32] for general $H(\xi)$.

######  Lemma 5.1.

If $z$ is a $C^{1}$ weak solution of (5.4), then we have

$\int_{\mathbb{R}^{N}}e^{z}dx\geq\left(\frac{N}{N-1}\right)^{N-1}N^{N}\kappa_{N}.$

If $z$ is a $C^{1}$ weak solution of (5.5), then we have

$\int_{\mathbb{R}^{N}_{+}(s_{0})}e^{z}dx\geq\left(\frac{N}{N-1}\right)^{N-1}N^{N}\kappa_{N}.$

###### Proof.

As in the proof of Theorem 1.1, we use a level set argument. First, we assume
$z$ is a solution of (5.4). Put

$\Omega_{t}=\\{x\in\mathbb{R}^{N}\ :\ z(x)>t\\},\quad\mu(t)=|\Omega_{t}|.$

Integration by parts on $\Omega_{t}$ leads to

$\displaystyle\int_{\Omega_{t}}e^{z}dx=-\int_{\Omega_{t}}Q_{N}zdx$
$\displaystyle=\int_{\partial\Omega_{t}}H^{N-1}(\nabla
z)(\nabla_{\xi}H)(\nabla z)\cdot\frac{\nabla z}{|\nabla z|}ds_{x}$
$\displaystyle=\int_{\partial\Omega_{t}}\frac{H^{N}(\nabla z)}{|\nabla
z|}ds_{x}.$

By the Finsler isoperimetric inequality (4.6) and Hölder’s inequality, we see

$\displaystyle N\kappa_{N}^{1/N}|\Omega_{t}|^{\frac{N-1}{N}}\leq
P_{H}(\Omega_{t},\mathbb{R}^{N})$
$\displaystyle=\int_{\partial\Omega_{t}}\frac{H(\nabla z)}{|\nabla z|}ds_{x}$
$\displaystyle\leq\left(\int_{\partial\Omega_{t}}\frac{H^{N}(\nabla
z)}{|\nabla
z|}ds_{x}\right)^{\frac{1}{N}}\left(\int_{\partial\Omega_{t}}\frac{ds_{x}}{|\nabla
z|}\right)^{\frac{N-1}{N}}$
$\displaystyle=\left(\int_{\Omega_{t}}e^{z}dx\right)^{\frac{1}{N}}\left(-\mu^{\prime}(t)\right)^{\frac{N-1}{N}},$

here we have used coarea formula

$\mu(t)=\int_{t}^{\infty}\int_{\\{x:z(x)=s\\}}\frac{ds_{x}}{|\nabla z|}ds.$

Thus we have

$\mu(t)\leq\left\\{\frac{1}{N\kappa_{N}^{1/N}}\left(\int_{\Omega_{t}}e^{z}dx\right)^{\frac{1}{N}}\left(-\mu^{\prime}(t)\right)^{\frac{N-1}{N}}\right\\}^{\frac{N}{N-1}}.$

Therefore, we obtain

$\displaystyle\int_{\mathbb{R}^{N}}e^{z}dx$
$\displaystyle=\int_{-\infty}^{\max z}e^{t}\mu(t)dt$
$\displaystyle\leq\left(\frac{1}{N\kappa_{N}^{1/N}}\right)^{\frac{N}{N-1}}\int_{-\infty}^{\max
z}e^{t}\left(\int_{\Omega_{t}}e^{z}dx\right)^{\frac{1}{N-1}}(-\mu^{\prime}(t))dt$
$\displaystyle=\left(\frac{1}{N\kappa_{N}^{1/N}}\right)^{\frac{N}{N-1}}\left(\frac{N-1}{N}\right)\int_{-\infty}^{\max
z}\frac{d}{dt}\left(\int_{\Omega_{t}}e^{z}dx\right)^{\frac{N}{N-1}}dt$
$\displaystyle=\left(\frac{1}{N\kappa_{N}^{1/N}}\right)^{\frac{N}{N-1}}\left(\frac{N-1}{N}\right)\left(\int_{\mathbb{R}^{N}}e^{z}dx\right)^{\frac{N}{N-1}},$

which implies

$\left(\frac{N}{N-1}\right)^{N-1}N^{N}\kappa_{N}\leq\int_{\mathbb{R}^{N}}e^{z}dx.$

The proof when $z$ is a solution to (5.5) is similar, since the boundary
condition $z=-\infty$ on $\partial\mathbb{R}^{N}_{+}(s_{0})$ assures that all
level sets of $z$ are confined in $\mathbb{R}^{N}_{+}(s_{0})$.

By the change of variables, we have

(5.6)
$p^{N-1}\int_{\Omega}u_{p}^{p+1}(y)dy=u_{p}^{N}(x_{p})\int_{\Omega_{p}}\left(1+\frac{z_{p}(x)}{p}\right)^{p+1}dx.$

Let us take $\limsup_{p\to\infty}$ of both sides of (5.6). Then we see

$\limsup_{n\to\infty}\text{LHS of
}\eqref{change}=\left(\frac{Ne}{N-1}\right)^{N-1}N^{N}\kappa_{N}$

by Lemma 3.3. On the other hand, Fatou’s lemma and Lemma 5.1 implies

$\displaystyle\limsup_{p\to\infty}\text{RHS of }\eqref{change}$
$\displaystyle\geq(\limsup_{p\to\infty}u_{p}(x_{p}))^{N}\times\begin{cases}\int_{\mathbb{R}^{N}}e^{z}dx\quad&\text{when
case (i)}\\\ \int_{\mathbb{R}^{N}_{+}(s_{0})}e^{z}dx\quad&\text{when case
(ii)}\end{cases}$
$\displaystyle\geq(\limsup_{p\to\infty}u_{p}(x_{p}))^{N}\left(\frac{N}{N-1}\right)^{N-1}N^{N}\kappa_{N}.$

Hence, we have

$e^{N-1}\geq(\limsup_{p\to\infty}\|u_{p}\|_{L^{\infty}(\Omega)})^{N}.$

which implies Theorem 1.2

##  6\. Proof of Theorem 1.3

In this section, we prove Theorem 1.3. Given any sequence $p_{n}$ of $p$ with
$p_{n}\to\infty$, let us recall (1.3) and (1.6) for $p=p_{n}$,
$u_{n}=u_{p_{n}}$.

$\displaystyle
v_{n}=\frac{u_{n}}{\lambda_{n}}=\frac{u_{n}}{(\int_{\Omega}u_{n}^{p_{n}}dx)^{\frac{1}{N-1}}},\quad\lambda_{n}=\left(\int_{\Omega}u_{n}^{p_{n}}dx\right)^{\frac{1}{N-1}},$
$\displaystyle f_{n}(x)=\frac{u_{n}^{p_{n}}}{\int_{\Omega}u_{n}^{p_{n}}dx},$
$\displaystyle
L_{0}=\limsup_{n\to\infty}\frac{p_{n}\left(\int_{\Omega}u_{n}^{p_{n}}dx\right)^{\frac{1}{N-1}}}{\left(\frac{N}{N-1}e^{\frac{N-1}{N}}\right)},\quad
L_{1}=d_{N}^{-\left(\frac{1}{N-1}\right)}L_{0}.$

Then $v_{n}$ is a weak solution of (1.4) for $p=p_{n}$. By Hölder’s inequality
and Theorem 1.1, we see

$p_{n}^{N-1}\int_{\Omega}u_{n}^{p_{n}}dx\leq
p_{n}^{N-1}\left(\int_{\Omega}u_{n}^{p_{n}+1}dx\right)^{\frac{p_{n}}{p_{n}+1}}|\Omega|^{\frac{1}{p_{n}+1}}\to\left(\frac{Ne}{N-1}\beta_{N}\right)^{N-1}$

as $n\to\infty$. This shows that

$L_{0}\leq e^{\frac{1}{N}}\beta_{N},\quad L_{1}\leq
e^{\frac{1}{N}}\beta_{N}d_{N}^{-(\frac{1}{N-1})}.$

First, we prove $S\neq\phi$ for any sequence $v_{n}=v_{p_{n}}$ of $v_{p}$ with
$p_{n}\to\infty$. Indeed, by Theorem 1.1, we have
$\|u_{n}\|_{L^{\infty}(\Omega)}\geq C_{1}>0$ for any $n\in\mathbb{N}$. Let
$x_{n}\in\Omega$ be a point such that
$u_{n}(x_{n})=\|u_{n}\|_{L^{\infty}(\Omega)}$, then

$v_{n}(x_{n})=\frac{u_{n}(x_{n})}{(\int_{\Omega}u_{n}^{p_{n}}dx)^{\frac{1}{N-1}}}\geq\frac{C_{1}}{(\int_{\Omega}u_{n}^{p_{n}}dx)^{\frac{1}{N-1}}}=\frac{C_{1}}{O(\frac{1}{p_{n}})}\to+\infty$

by Lemma 3.3. This implies that any accumulation point of $\\{x_{n}\\}$ is
contained in $S$ and hence $S\neq\phi$.

Next, as in [6], [21], [22], we define $(L,\delta)$-regular set and
$(L,\delta)$-irregular set of a sequence $\\{u_{n}\\}$. Since

$f_{n}=\frac{u_{n}^{p_{n}}}{\int_{\Omega}u_{n}^{p_{n}}dx}\in
L^{1}(\Omega),\quad f_{n}\geq 0,\quad\int_{\Omega}f_{n}dx=1,$

there exists a subsequence (still denoted by $n$) such that

$f_{n}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu,\quad\mu(\Omega)\leq 1$

in the sense of Radon measures of $\Omega$, where $\mu$ is a nonnegative Radon
measure.

Given $L>0$ and $\delta>0$, we call a point $x_{0}\in\Omega$ a
$(L,\delta)$-regular point of $\\{u_{n}\\}$ if there exists $\varphi\in
C_{0}(\Omega),0\leq\varphi\leq 1$ with $\varphi\equiv 1$ near $x_{0}$ such
that

$\int_{\Omega}\varphi d\mu<\left(\frac{\beta_{N}}{L+3\delta}\right)^{N-1}$

where $\beta_{N}=N(N\kappa_{N})^{\frac{1}{N-1}}$ is as in Theorem 2.2. We put

$\displaystyle R_{L}(\delta)=\\{x_{0}\in\Omega:x_{0}\;\mbox{is
a}\;(L,\delta)\text{-regular point}\\},$
$\displaystyle\Sigma_{L}(\delta)=\Omega\setminus R_{L}(\delta).$

We call a point in $\Sigma_{L}(\delta)$ an $(L,\delta)$-irregular point of the
sequence $\\{u_{n}\\}$. Note that $(L,\delta)$-regular, or
$(L,\delta)$-irregular points are automatically interior points of $\Omega$.
Also note that if $x_{0}\in\Sigma_{L}(\delta)$, then we have

(6.1) $\mu(\\{x_{0}\\})\geq\left(\frac{\beta_{N}}{L+3\delta}\right)^{N-1}.$

Since

$1\geq\mu(\Omega)\geq\left(\frac{\beta_{N}}{L+3\delta}\right)^{N-1}\sharp\Sigma_{L}(\delta)$

by (6.1), we see that $\Sigma_{L}(\delta)$ is a finite set for any $L>0$ and
$\delta>0$.

Next Lemma is the key to analyze the interior blow-up set $S\cap\Omega$.

######  Lemma 6.1.

(smallness of $\mu$ implies boundedness) Let $x_{0}$ be a
$(L_{1},\delta)$-regular point of a sequence $\\{u_{n}\\}$ where $L_{1}$ is
defined in (1.6). Then $\\{v_{n}\\}$ is bounded in
$L^{\infty}(B_{R_{0}}(x_{0}))$ for some $R_{0}>0$.

###### Proof.

Key point in the proof is to get the following pointwise estimate

(6.2)
$f_{n}(x)<\exp\left((L_{1}+\delta/2)d_{N}^{\frac{1}{N-1}}v_{n}(x)\right),\quad
x\in\Omega.$

In checking (6.2), we use the elementary inequality

(6.3) $\frac{\log x}{x}\leq\frac{\log y}{y}\quad\text{for any}\;0<x\leq y\leq
e.$

Let

$\alpha_{n}=\frac{\|u_{n}\|_{L^{\infty}(\Omega)}}{\left(\int_{\Omega}u_{n}^{p_{n}}dx\right)^{\frac{1}{p_{n}}}}=\frac{\|u_{n}\|_{L^{\infty}(\Omega)}}{\lambda_{n}^{\frac{N-1}{p_{n}}}},$

and recall that $\lambda_{n}=O\left(\frac{1}{p_{n}}\right)$ by Corollary 4.1,
so
$\lambda_{n}^{\frac{N-1}{p_{n}}}=O\left(\frac{1}{p_{n}}\right)^{\frac{N-1}{p_{n}}}\to
1$ as $n\to\infty$. Thus we have

$\limsup_{n\to\infty}\alpha_{n}=\limsup_{n\to\infty}\|u_{n}\|_{L^{\infty}(\Omega)}\leq
e^{\frac{N-1}{N}}$

by Theorem 1.2. From this, we see that for any small $\varepsilon^{\prime}>0$,

$\frac{u_{n}(x)}{\lambda_{n}^{\frac{N-1}{p_{n}}}}\leq\alpha_{n}\leq
e^{\frac{N-1}{N}}+\varepsilon^{\prime}<e$

holds for any $x\in\Omega$ and for large $n$. Therefore by (6.3), we have for
fixed small $\varepsilon>0$

$\frac{\log\left(\frac{u_{n}(x)}{\lambda_{n}^{\frac{N-1}{p_{n}}}}\right)}{\frac{u_{n}(x)}{\lambda_{n}^{\frac{N-1}{p_{n}}}}}\leq\frac{\log\alpha_{n}}{\alpha_{n}}\leq\left(\frac{N-1}{N}\right)\frac{1}{e^{\frac{N-1}{N}}}+\varepsilon$

for large $n$. Hence

$\displaystyle\log
f_{n}(x)=p_{n}\log\frac{u_{n}(x)}{\lambda_{n}^{\frac{N-1}{p_{n}}}}$
$\displaystyle\leq
p_{n}\left(\frac{u_{n}(x)}{\lambda_{n}^{\frac{N-1}{p_{n}}}}\right)\left(\frac{N-1}{N}\frac{1}{e^{\frac{N-1}{N}}}+\varepsilon\right)$
$\displaystyle=p_{n}\lambda_{n}\left(\frac{N-1}{N}e^{-\frac{N-1}{N}}+\varepsilon\right)\frac{v_{n}(x)}{\lambda_{n}^{\frac{N-1}{p_{n}}}}$
$\displaystyle\leq\left(\frac{N}{N-1}e^{\frac{N-1}{N}}L_{1}d_{N}^{\frac{1}{N-1}}+\varepsilon\right)\left(\frac{N-1}{N}e^{-\frac{N-1}{N}}+2\varepsilon\right)v_{n}(x),$

here we have used $\lim_{n\to\infty}\lambda_{n}^{\frac{N-1}{p_{n}}}=1$ and

$p_{n}\lambda_{n}\leq\frac{N}{N-1}e^{\frac{N-1}{N}}L_{1}d_{N}^{\frac{1}{N-1}}+\varepsilon$

for large $n$ by the definition of $L_{1}$. Therefore

$\log f_{n}(x)\leq\left((L_{1}+\delta/2)d_{N}^{\frac{1}{N-1}}\right)v_{n}(x)$

holds if we choose $\varepsilon>0$ small enough. This proves the pointwise
estimate (6.2).

Next, by the use of Brezis-Merle theory for the Finsler $N$-Laplacian, we
obtain the integral estimate

(6.4)
$\int_{B_{R_{1}/2}(x_{0})}\exp\left((L_{1}+\delta)d_{N}^{\frac{1}{N-1}}v_{n}(x)\right)dx\leq
C$

for some $R_{1}>0$ small and $C>0$ independent of $n$, here $x_{0}$ is a
$(L_{1},\delta)$-regular point.

Indeed, by the definition of $(L_{1},\delta)$-regular point, we can find
$R_{1}>0$ such that

$\int_{B_{R_{1}}(x_{0})}f_{n}dx\leq\left(\frac{\beta_{N}}{L_{1}+2\delta}\right)^{N-1}.$

Also by Theorem 2.5 (i) and the fact that $\|f_{n}\|_{L^{1}(\Omega)}=1$, we
have

$\int_{\Omega}\exp\left((\beta_{N}-\varepsilon)v_{n}(x)\right)dx\leq\frac{\beta_{N}}{\varepsilon}|\Omega|$

for any $\varepsilon\in(0,\beta_{N})$. From this, we obtain

(6.5) $\|v_{n}\|_{L^{N}(\Omega)}\leq C$

where $C>0$ is independent of $n$. Next, let $\phi_{n}$ be a weak solution of

$-Q_{N}\phi_{n}=0\quad\text{in}\,B_{R_{1}}(x_{0})\quad\phi_{n}=v_{n}\quad\text{on}\,\partial
B_{R_{1}}(x_{0}).$

Then by Theorem 2.5 (2) and the fact that
$\|f_{n}\|_{L^{1}(B_{R_{1}}(x_{0}))}^{\frac{1}{N-1}}<\frac{\beta_{N}}{L_{1}+2\delta}$,
we have

(6.6)
$\int_{\Omega}\exp\left((L_{1}+\delta)d_{N}^{\frac{1}{N-1}}|v_{n}(x)-\phi_{n}(x)|\right)dx\leq
C$

if we choose $\varepsilon\in(0,\beta_{N})$ sufficiently small. By the
comparison principle for the Finsler $N$-Laplacian (see [33] Theorem 3.2) and
Serrin’s estimates Theorem 2.4 (i), we have

$\|\phi_{n}\|_{L^{\infty}(B_{R_{1}/2}(x_{0}))}\leq\|v_{n}\|_{L^{\infty}(B_{R_{1}/2}(x_{0}))}\leq
C\|v_{n}\|_{L^{N}(B_{R_{1}}(x_{0}))}\leq C$

where we have used (6.5). Combining this with (6.6), we obtain the desired
integral estimate (6.4).

Comparing (6.4) and (6.2), we see that $f_{n}$ is bounded uniformly in $n$ in
$L^{q}(B_{R_{1}/2}(x_{0}))$ where $q=\frac{L_{1}+\delta}{L_{1}+\delta/2}>1$.
Therefore, Serrin’s regularity estimate Theorem 2.4 (i) again implies that

$\|v_{n}\|_{L^{\infty}(B_{R_{1}/4}(x_{0}))}\leq C$

independent of $n$. Taking $R_{0}=R_{1}/4$ ends the proof of Lemma 6.1.

We know that $\Sigma_{L_{1}}(\delta)$ is a set of finite points, all of those
are interior of $\Omega$. From Lemma 6.1, we obtain
$S\cap\Omega=\Sigma_{L_{1}}(\delta)$ for any $\delta>0$ and

$1\geq\mu(\Omega)\geq\left(\frac{\beta_{N}}{L_{1}+3\delta}\right)^{N-1}\sharp(\Sigma_{L_{1}}(\delta))=\left(\frac{\beta_{N}}{L_{1}+3\delta}\right)^{N-1}\sharp(S\cap\Omega).$

Hence

$\sharp(S\cap\Omega)\leq\left(\frac{L_{1}+3\delta}{\beta_{N}}\right)^{N-1}\leq\left(\frac{e^{\frac{1}{N}}\beta_{N}d_{N}^{-(\frac{1}{N-1})}+3\delta}{\beta_{N}}\right)^{N-1}.$

Taking a limit $\delta\to 0$, we have

$\sharp(S\cap\Omega)\leq e^{\frac{N-1}{N}}d_{N}^{-1}$

This proves the first part of Theorem 1.3.

If $x_{0}\in S\cap\Omega=\Sigma_{L_{1}}(\delta)$, then for any $R>0$, we have

(6.7) $\lim_{n\to\infty}\|v_{n}\|_{L^{\infty}(B_{R}(x_{0}))}=+\infty.$

Indeed, if for some $R>0$, assume there exists $C>0$ independent of $n$ such
that $\|v_{n}\|_{L^{\infty}(B_{R}(x_{0}))}\leq C$ for all large $n$. Then

$f_{n}=\frac{v_{n}^{p_{n}}}{\lambda_{n}^{N-1-p_{n}}}\leq
C^{p_{n}}O\left(\frac{1}{p_{n}}\right)^{p_{n}-(N-1)}\to 0\quad(n\to\infty)$

uniformly on $B_{R}(x_{0})$. This implies $x_{0}$ is a
$(L_{1},\delta)$-regular point, which is absurd. The same kind of argument
leads to that the limit measure $\mu$ is atomic and of the form

$\mu=\sum_{i=1}^{k}\gamma_{i}\delta_{x_{i}}$

where $S\cap\Omega=\\{x_{1},\cdots,x_{k}\\}$. Since $\mu(\Omega)\leq 1$, we
have $\sum_{i=1}^{k}\gamma_{i}\leq 1$ and

$\gamma_{i}\geq(\frac{\beta_{N}}{L_{1}})^{N-1}$

for all $i=1,\cdots,k$ by letting $\delta\to 0$ in (6.1) with $L=L_{1}$. This
proves Theorem 1.3 (i).

On any compact sets in $\Omega\setminus(S\cap\Omega)$, $\\{v_{n}\\}$ is
uniformly bounded. Then by Serrin’s and Tolksdorf’s regularity estimate,
$\\{v_{n}\\}$ is also bounded in
$C^{1,\alpha}_{loc}(\Omega\setminus(S\cap\Omega))$ for some $\alpha\in(0,1)$.
By Ascoli-Arzelá theorem, we have a subsequence and a function $G$ such that
$v_{n}\to G$ in $C^{1}_{loc}(\Omega\setminus(S\cap\Omega))$. That this $G$
satisfies Theorem 1.3 (ii) is clear.

Finally, since $\lambda_{n}=O(\frac{1}{p_{n}})$ as $n\to\infty$ and
$v_{n}(x)=\frac{u_{n}(x)}{\lambda_{n}}$ is uniformly bounded in
$L^{\infty}_{loc}(\Omega\setminus(S\cap\Omega))$, we easily see that Theorem
1.3 (iii) holds.

Thus all the proof of Theorem 1.3 has been completed.

##  7\. Proof of Theorem 1.4

In this section, we prove Theorem 1.4.

###### Proof.

Assume the contrary that $x_{0}\in\partial\Omega$, where $x_{0}$ is the unique
blow-up point of a sequence $v_{n}=v_{p_{n}}$ with $p_{n}\to+\infty$ as
$n\to\infty$. For $R>0$ small, we may use the Pohozaev identity Theorem 2.6 on
$\Omega\cap B_{R}(x_{0})$, with the aid of Theorem 2.4:

(7.1) $\displaystyle\frac{N}{p_{n}+1}\int_{\Omega\cap
B_{R}(x_{0})}u_{n}^{p_{n}+1}dx=\int_{\partial(\Omega\cap
B_{R}(x_{0}))}\frac{u_{n}^{p_{n}+1}}{p_{n}+1}(x-y)\cdot\nu(x)ds_{x}$
$\displaystyle-\frac{1}{N}\int_{\partial(\Omega\cap B_{R}(x_{0}))}H^{N}(\nabla
u_{n})(x-y)\cdot\nu(x)ds_{x}$ $\displaystyle+\int_{\partial(\Omega\cap
B_{R}(x_{0}))}\left(H^{N-1}(\nabla u_{n})(\nabla_{\xi}H)(\nabla
u_{n})\cdot\nu(x)\right)(x-y)\cdot\nu(x)ds_{x}.$

In order to remove the integral terms involving $\partial\Omega$, we use a
trick in [25]. Define

$\rho_{n}=\frac{\int_{\partial\Omega\cap B_{R}(x_{0})}H^{N}(\nabla
u_{n})(x-x_{0})\cdot\nu(x)ds_{x}}{\int_{\partial\Omega\cap
B_{R}(x_{0})}H^{N}(\nabla u_{n})\nu(x_{0})\cdot\nu(x)ds_{x}}$

and put $y_{n}=x_{0}+\rho_{n}\nu(x_{0})$. We assume $R>0$ so small such that
$1/2\leq\nu(x_{0})\cdot\nu(x)\leq 1$ for $x\in\partial\Omega\cap
B_{R}(x_{0})$. Then we have that $\rho_{n}\leq 2R$. By the definition of
$y_{n}$ and $\rho_{n}$, we see that

$\int_{\partial\Omega\cap B_{R}(x_{0})}H^{N}(\nabla
u_{n})(x-y_{n})\cdot\nu(x)ds_{x}\equiv 0$

for all $n\in\mathbb{N}$. Also since $u_{n}=0$ on $\partial\Omega$ and
$u_{n}>0$ in $\Omega$, we see $\nu(x)=-\frac{\nabla u_{n}(x)}{|\nabla
u_{n}(x)|}$. By using these, we see (7.1) with $y=y_{n}$ becomes

(7.2) $\displaystyle\frac{N}{p_{n}+1}\int_{\Omega\cap
B_{R}(x_{0})}u_{n}^{p_{n}+1}dx=\frac{1}{p_{n}+1}\int_{\Omega\cap\partial
B_{R}(x_{0})}u_{n}^{p_{n}+1}(x-y_{n})\cdot\nu(x)ds_{x}$
$\displaystyle-\frac{1}{N}\int_{\Omega\cap\partial B_{R}(x_{0})}H^{N}(\nabla
u_{n})(x-y_{n})\cdot\nu(x)ds_{x}$ $\displaystyle+\int_{\Omega\cap\partial
B_{R}(x_{0}))}\left(H^{N-1}(\nabla u_{n})(\nabla_{\xi}H)(\nabla
u_{n})\cdot\nu(x)\right)(x-y_{n})\cdot\nabla u_{n}(x)ds_{x}.$

Multiplying $(\frac{1}{\lambda_{n}})^{N}$ to both sides of (7.2) and recalling
$v_{n}=\frac{u_{n}}{\lambda_{n}}$, we have

(7.3)
$\displaystyle\frac{N}{p_{n}+1}\left(\frac{1}{\lambda_{n}}\right)^{N}\int_{\Omega\cap
B_{R}(x_{0})}u_{n}^{p_{n}+1}dx$
$\displaystyle=\frac{1}{p_{n}+1}\left(\frac{1}{\lambda_{n}}\right)^{N}\int_{\Omega\cap\partial
B_{R}(x_{0})}u_{n}^{p_{n}+1}(x-y_{n})\cdot\nu(x)ds_{x}$
$\displaystyle-\frac{1}{N}\int_{\Omega\cap\partial B_{R}(x_{0})}H^{N}(\nabla
v_{n})(x-y_{n})\cdot\nu(x)ds_{x}$ $\displaystyle+\int_{\Omega\cap\partial
B_{R}(x_{0}))}\left(H^{N-1}(\nabla v_{n})(\nabla_{\xi}H)(\nabla
v_{n})\cdot\nu(x)\right)(x-y_{n})\cdot\nabla v_{n}(x)ds_{x}$
$\displaystyle=I+II+III.$

We estimate the terms $I,II,III$ on the right-hand side of (7.3) as follows:

$\displaystyle|I|=\frac{1}{p_{n}+1}\left(\frac{1}{\lambda_{n}}\right)^{N}\big{|}\int_{\Omega\cap\partial
B_{R}(x_{0})}u_{n}^{p_{n}+1}(x-y_{n})\cdot\nu(x)ds_{x}\big{|}$
$\displaystyle\leq\frac{O(p_{n}^{N})}{p_{n}^{N-1}(p_{n}+1)}\|p_{n}^{N-1}u_{n}^{p_{n}+1}\|_{L^{\infty}(\Omega\cap\partial
B_{R}(x_{0}))}\int_{\Omega\cap\partial
B_{R}(x_{0})}|(x-y_{n})\cdot\nu(x)|ds_{x}$
$\displaystyle=\frac{O(p_{n}^{N})}{p_{n}^{N-1}(p_{n}+1)}\|p_{n}^{N-1}u_{n}^{p_{n}+1}\|_{L^{\infty}(\Omega\cap\partial
B_{R}(x_{0}))}O(R^{N-1}).$

We note that since $S\cap\Omega=\phi$ by assumption,

$f_{n}=\frac{u_{n}^{p_{n}}}{\lambda_{n}^{N-1}}\to 0$

uniformly on compact sets in $\Omega$ and

$p_{n}^{N-1}u_{n}^{p_{n}+1}(x)\leq\|u_{n}\|_{L^{\infty}(\Omega)}p_{n}^{N-1}u_{n}^{p_{n}}(x)\leq
C\frac{u_{n}^{p_{n}}(x)}{\lambda_{n}^{N-1}}\leq Cf_{n}(x)$

by Theorem 1.1 and the fact that $\lambda_{n}=O(\frac{1}{p_{n}})$ as
$n\to\infty$. Thus we have

$\|p_{n}^{N-1}u_{n}^{p_{n}+1}\|_{L^{\infty}(\Omega\cap\partial
B_{R}(x_{0}))}\to 0\quad\text{as}\quad n\to\infty$

and thus

$\lim_{R\to 0}\lim_{n\to\infty}|I|=0.$

Also, by Theorem 1.3 (ii), we have $v_{n}\to G$ in
$C^{1,\alpha}_{loc}(\Omega\setminus(S\cap\Omega))$. Thus we have $H^{N}(\nabla
v_{n})=O(1)$ on $\Omega\cap\partial B_{R}(x_{0})$, which implies

$\displaystyle|II|$ $\displaystyle=\frac{1}{N}\Big{|}\int_{\Omega\cap\partial
B_{R}(x_{0})}H^{N}(\nabla v_{n})(x-y_{n})\cdot\nu(x)ds_{x}\Big{|}$
$\displaystyle\leq O(1)\int_{\Omega\cap\partial
B_{R}(x_{0})}|(x-y_{n})\cdot\nu(x)|ds_{x}\leq O(1)O(R^{N-1}),$
$\displaystyle|III|$ $\displaystyle=\Big{|}\int_{\Omega\cap\partial
B_{R}(x_{0}))}\left(H^{N-1}(\nabla v_{n})(\nabla_{\xi}H)(\nabla
v_{n})\cdot\nu(x)\right)(x-y_{n})\cdot\nabla v_{n}(x)ds_{x}\Big{|}$
$\displaystyle\leq O(1)\int_{\Omega\cap\partial
B_{R}(x_{0})}|(x-y_{n})\cdot\nu(x)|ds_{x}\leq O(1)O(R^{N-1}).$

Therefore we have

$\lim_{R\to 0}\lim_{n\to\infty}|II|=\lim_{R\to 0}\lim_{n\to\infty}|III|=0.$

From these, we obtain

(7.4) $\lim_{R\to 0}\lim_{n\to\infty}(\text{RHS of
\eqref{local_Pohozaev3}})=0.$

On the other hand, recall

$\displaystyle z_{n}(x)$
$\displaystyle=\frac{p_{n}}{u_{n}(x_{n})}\left(u_{n}(\varepsilon_{n}x+x_{n})-u_{n}(x_{n})\right),$
$\displaystyle x\in\Omega_{R,n}=\frac{(\Omega\cap
B_{R}(x_{0}))-x_{n}}{\varepsilon_{n}},$

where $\varepsilon_{n}^{N}p_{n}^{N-1}u_{n}(x_{n})^{p_{n}+1-N}\equiv 1$. Then
we see from Fatou’s lemma, Theorem 1.1, and Lemma 5.1, that

$\displaystyle\lim_{R\to 0}\lim_{n\to\infty}\int_{\Omega\cap
B_{R}(x_{0}))}p_{n}^{N-1}u_{n}^{p_{n}+1}(y)dy$ $\displaystyle=\lim_{R\to
0}\lim_{n\to\infty}u_{n}(x_{n})^{N}\int_{\Omega_{R,n}}\left(1+\frac{z_{n}(x)}{p_{n}}\right)^{p_{n}+1}dx$
$\displaystyle\geq C_{1}^{N}\int_{U}e^{z}dx\geq
C_{1}^{N}\left(\frac{N}{N-1}\right)^{N-1}N^{N}\kappa_{N}$

where $u=\mathbb{R}^{N}$ or $\mathbb{R}^{N}_{+}(s_{0})$ for some $s_{0}>0$
according to the cases $\frac{{\rm
dist}(x_{n},\partial\Omega_{R,n})}{\varepsilon_{n}}\to+\infty$ or $\frac{{\rm
dist}(x_{n},\partial\Omega_{R,n})}{\varepsilon_{n}}\to s_{0}$. Note that our
assumption $\sharp S=1$ assures that we can choose $x_{n}$ as a maximum points
of $u_{n}$. From this and the fact that $\lambda_{n}=O(\frac{1}{p_{n}})$ as
$n\to\infty$, we have

(7.5) $\lim_{R\to 0}\lim_{n\to\infty}(\text{LHS of
\eqref{local_Pohozaev3}})\geq C>0$

for some positive constant $C>0$ independent of $n$.

Clearly (7.5) contradicts to (7.4), and we conclude that
$x_{0}\not\in\partial\Omega$.

Finally, as a corollary, we prove the following.

######  Corollary 7.1.

Let $R>0$ and let $\\{u_{p}\\}$ be a sequence of least energy solutions to

(7.6) $\begin{cases}&-Q_{N}u_{p}=u_{p}^{p}\quad\mbox{in}\;\mathcal{W}_{R},\\\
&u_{p}>0\quad\mbox{in}\;\mathcal{W}_{R},\\\
&u_{p}=0\quad\mbox{on}\;\partial\mathcal{W}_{R}\end{cases}$

where $\mathcal{W}_{R}=\\{x\in\mathbb{R}^{N}\,:\,H^{0}(x)<R\\}$. Then the
blow-up set $S$ of $v_{p}$ satisfies $S\cap\mathcal{W}_{R}=\\{0\\}$, and

$\displaystyle u_{p}\to G(\cdot,0)\quad\text{in
}C^{1}_{loc}(\mathcal{W}_{R}\setminus\\{0\\})$

where $G$ is the unique Green function on $\mathcal{W}_{R}$ obtained in
Theorem 2.3, and

$\displaystyle
f_{p}=\frac{u_{p}^{p}}{\int_{\mathcal{W}_{R}}u_{p}^{p}dx}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\delta_{0}$

in the sense of Radon measures on $\mathcal{W}_{R}$, along the full sequence.

###### Proof.

The usual method of moving plane to prove the symmetry of solutions is not
applicable in the anisotropic situation. However, we can use Theorem 4.1 in
[5] under the convexity and $C^{1}$-assumption of the map $\xi\mapsto
H^{N}(\xi)$. (Note that the key point of the proof of Theorem 4.1 in [5] is
the Pohozaev identity Theorem 4.2 in [5] for $C^{1}(\overline{\Omega})$-weak
solutions, which is valid by the above assumptions). Thus we assure that any
positive solution $u_{p}$ to (7.6) is Finsler-radial, that is, all level sets
of $u_{p}$ are homothetic to $\mathcal{W}_{R}$ for any $p>1$. Let $S$ be the
blow-up set of $v_{p}$. Then we see that $S\cap\mathcal{W}_{R}=\\{0\\}$.
Indeed, if there were a point $x_{0}\in S\cap\mathcal{W}_{R}$, then all points
on the level set of $u_{p}$ passing through $x_{0}$ must be blow-up points of
$v_{p}$, which contradicts to the fact that $\sharp(S\cap\mathcal{W}_{R})$ is
finite. Thus by Theorem 1.3, we see

$\displaystyle v_{p}\to G(\cdot,0)\quad\text{in
}C^{1}_{loc}(\mathcal{W}_{R}\setminus\\{0\\})$

for some function $G$ along a subsequence. The limit function must be the
unique Green function constructed in Theorem 2.3, and by the uniqueness, the
convergence is true for the full sequence.

Acknowledgments.

This work was partly supported by Osaka City University Advanced Mathematical
Institute MEXT Joint Usage / Research Center on Mathematics and Theoretical
Physics JPMXP0619217849. The second author (F.T.) was supported by JSPS
KAKENHI Grant-in-Aid for Scientific Research (B), JP19H01800, and JSPS Grant-
in-Aid for Scientific Research (S), JP19H05597.

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# Comparison of SMC and OMC results in determining the ground-state and meta-
stable states solutions for UO2 in DFT+U method

Mahmoud Payami  Email<EMAIL_ADDRESS>School of Physics & Accelerators,
Nuclear Science and Technology Research Institute, AEOI, P. O. Box 14395-836,
Tehran, Iran

###### Abstract

Correct prediction of the behavior of UO2 crystal, which is an antiferromagnet
system with strongly-correlated electrons, is possible by using a modified
density functional theory, the DFT+U method. In the context of DFT+U, the
energy of crystal turns out to be a function with several local minima, the
so-called meta-stable states, and the lowest energy state amongst them is
identified as the ground state. OMC was a method that were used in DFT+U to
determine the ground state. Recently the SMC method was proposed which also
revealed the multi-minima structure of energy in the DFT+U approach and led to
results in good agreement with experiment. In this work, we compare the SMC
and OMC results and show that although the ground states of the two methods
have similar energies and geometries, the electronic structures have
significant differences. Moreover, the energy band gap of SMC is in better
agreement with experiments.

##  I Introduction

Uranium dioxide is a common fuel used in nuclear power reactors. It has a
3k-order anti-ferromagnetic (AFM) crystal structure at temperatures less than
$30~{}$K and para-magnetic structure at higher temperatures[1, 2]. From
earlier experimental results [3] it was concluded that the uranium and oxygen
atoms occupied respectively the octahedral (4a) and tetrahedral (8c) symmetry
positions of cubic space group $Fm\bar{3}m$ (No. 225) with lattice constant of
5.47Å, which is shown in Fig. 1(a). This space group can be represented by a
simple tetragonal unit cell with 6 atoms as shown in Fig. 1(b). However, a
later XRD experiment[4] indicated that UO2 crystallizes with a less symmetric
cubic space group $Pa\bar{3}$ (No. 205) with oxygen atoms slightly displaced
inside the cube.

(a)

(b)

Figure 1: (a)- UO2 crystal structure with cubic space group $Fm\bar{3}m$ (No.
225) and lattice constant of 5.47Å; (b)- description by a simple tetragonal
crystal structure with six atoms. Gray and small red balls represent uranium
and oxygen atoms, respectively.

The use of ordinary approximations in density-functional theory (DFT)[5, 6]
description of the system leads to incorrect metallic behavior while it is
experimentally found[7] to be an insulator with a gap of 2.10 eV. The
incorrect metallic prediction arises from the partially-filled ”localized”
$5f$ or $6d$ valence electrons in uranium atoms that are treated in the same
footing as other delocalized ones in the atom. One of the ways to overcome
this problem is the method of DFT+U [8, 9, 10, 11] in which the DFT
contributions of Hubbard orbitals are subtracted from the DFT energy
($E_{dc}$), and a better contribution, which is borrowed from Hubbard model
($E_{Hub}$), is added to DFT energy, leading to the DFT+U energy functional:

$E_{DFT+U}=E_{DFT}-E_{dc}+E_{Hub}.$ (1)

In Eq. (1), the first term in the right hand side is the LDA/GGA [6, 12] total
energy of the system. $E_{Hub}$ depends on the elements of the occupation
matrices of highly correlated $f$ and $d$ electrons of uranium atoms. It turns
out that $E_{DFT+U}$ is a functional having several local minima, the so-
called meta-stable states. To determine the ground state (GS) of the system,
one has to carefully determine all such local minimum states and then identify
the lowest energy state as the GS of the system. Two of the methods that
determine the meta-stable states are called the occupation-matrix control [10]
(OMC) and the starting magnetization control [21] (SMC). In this work, we
briefly describe these methods and by comparing the results primarily show
that the number of metastable states depends on the method and the
computational details. Then, it is shown that even though the GS states
determined by these two methods have similar energies, the electronic
structure of the two GS’s have significant differences. Moreover, the energy
gap of the SMC-GS is in better agreement with experiment.

The organization of this paper is as follows. In Section II, we present the
computational details; in Section III the calculated results are presented and
discussed; Section IV concludes this work.

##  II Computational details

All DFT+U calculations are based on the solution of the KS equations using the
Quantum-ESPRESSO code package [13, 14]. For the atoms U and O, we have
employed the ultra-soft pseudo-potentials (USPP) generated by the atomic code,
using the generation inputs (with small modifications for more desired
results) from the pslibrary [15], at https://github.com/dalcorso/pslibrary.
For the USPP generation, we have used the valence configurations of
U($6s^{2},\,6p^{6},\,7s^{2},\,7p^{0},\,6d^{1},\,5f^{3}$) and
O($2s^{2},\,2p^{4}$); and to take into account the dominant relativistic
effects of the electrons [16], we have adopted the scalar-relativistic (SR)
method[17]. For the exchange-correlation, we have chosen the Perdew-Zunger
[18] (PZ) LDA, which gives excellent geometric properties whenever the
projection on to Hubbard orbitals are chosen to be atomic ones that are not
orthonormalized. Performing convergency tests, the appropriate kinetic energy
cutoffs for the plane-wave expansions were chosen as 90 and 720 Ry for the
wavefunctions and densities, respectively. To avoid the self-consistency
problems, the Methfessel-Paxton smearing method [19] for the occupations with
a width of 0.01 Ry is used. For the Brillouin-zone integrations in geometry
optimizations, a $8\times 8\times 6$ grid were used; while for density-of-
states (DOS) calculations, we have used a denser grid of $10\times 10\times 7$
in reciprocal space and ”tetrahedron” method [20] for the occupations. In
DFT+U calculations, we have used the optimum value of 4.53 eV for Hubbard-U
parameter which gives excellent results for the geometric properties. All
geometries were fully optimized for total residual pressures on unit cells to
within 0.5 kbar, and residual forces on atoms to within 10-3 mRy/a.u.

###  II.1 OMC method

In this method, the starting magnetization for oxygen atoms are set to zero
values and for U atoms they are set to $\pm 0.5$ to make anti-ferromagnetic
(AFM) configuration along the $z$ direction. Moreover, the initial values for
the diagonal elements of occupation matrices in Hubbard correction are set to
zero and unity ($n_{\sigma}=0,1;\;\;\sigma=\uparrow,\downarrow$). Since in the
present work we apply Hubbard corrections to only $5f$ localized orbitals, the
dimension of the occupation matrices are equal to 7. On the other hand, since
in our employed U-atom pseudo-potential the $5f$ orbital is occupied by 3
electrons, then we can have $C_{3}^{7}=35$ different ways for occupying the
diagonal elements of $7\times 7$ matrix by 3 electrons: [1110000], [1101000],
[1100100], $\cdots$, [0001011], [0000111]. Among these occupation
configurations the ones which lead to the lowest energy are taken as the
initial occupations for GS. All others lead to metastable states. On the other
hand, some researchers have argued that only two $f$ orbitals of U atoms
should be filled[10], and therefore ending up with a $C_{2}^{7}=21$ different
ways for occupying the diagonal elements: [1100000], [1001000], [1000100],
$\cdots$, [0000101], [0000011]. We have considered both cases in our
calculations.

###  II.2 SMC method

In the SMC method [21], the starting magnetizations for the U atoms are set to
$\pm 0.5$ as in the OMC method, but the starting magnetization for the
O-atoms, $\zeta$, are considered as new degrees of freedom. For the initial
configurations of electrons, different values for starting magnetization in
the interval $-1.0\leq\zeta\leq+1.0$ with step $\Delta\zeta=0.1$ were
considered. This new degrees of freedom may be taken into account in two ways:
(a)-All-equivalent O atoms, and (b)-Two inequivalent O atoms. In the former
case, all oxygen atoms in the unit cell take the same values for $\zeta$, and
in the latter case, since there are two inequivalent uranium atoms with
magnetizations $\pm 0.5$, we may consider two independent degrees of freedom
for oxygen atoms which are denoted by $\zeta_{1}$ and $\zeta_{2}$. The
situation is shown in Fig. 2. In this method the number of SCF calculations,
441, is much more than that in OMC which equals at most to 35.

(a)

(b)

Figure 2: (a)- All-equivalent oxygen atoms scheme; and (b)-two-inequivalent
oxygen atoms scheme. In the all-equivalent oxygen atoms model, all oxygen
atoms are treated as the same type, O1, and assume the same starting
magnetizations within the SMC method; while in two-inequivalent oxygen atoms
model, the O1 and O2 oxygen atoms are treated as different types and assume
independent values for the starting magnetizations in the SMC method [21].

##  III Results and Discussions

###  III.1 OMC results

####  III.1.1 $C_{3}^{7}$ occupations

The self-consistent solutions in the OMC method with $C_{3}^{7}$ occupations
resulted in 12 local minima, the lowest energy one is identified as the GS.
The results are summarized in Table 1.

Table 1: GS and meta-stable (MS) states’ properties in the OMC-$C^{7}_{3}$ method. The energies are in Ry/(formula unit) and are compared to the GS. Equilibrium lattice constants are in $\AA$, total magnetizations are in Bohr-magneton/(formula unit), and energy gap is in eV.  State | initial occupations | $\Delta E$ |  $a$ ($c$) | Tot. mag. | $E_{gap}$
---|---|---|---|---|---
GS | [1110000],[0110010],[0110001] | 0.00000 | 5.4656 (5.4787) | 0.00 | 2.91
MS1 | [1000011],[0100011],[0010011] | 0.00464 | 5.4888 (5.4327) | 0.00 | 2.22
MS2 | [1001100],[0101100],[0011100],[0001110],[0001101] | 0.00838 | 5.4755 (5.4526) | 0.00 | 2.15
MS3 | [1001010],[1001001],[0101010],[0011010],[0001011] | 0.05790 | 5.4354 (5.4628) | 0.00 | metal
MS4 | [1010010] | 0.06170 | 5.4479 (5.4543) | 0.00 | metal
MS5 | [1000110],[1000101],[0100110],[0010110] | 0.06203 | 5.4362 (5.4694) | 0.00 | metal
MS6 | [0111000],[1101000],[0101001],[0011001],[1011000] | 0.06206 | 5.4444 (5.4882) | 0.00 | metal
MS7 | [0100101],[1100100],[0010101],[1010100] | 0.06865 | 5.4588 (5.4571) | 0.00 | metal
MS8 | [0000111] | 0.06979 | 5.4446 (5.4418) | 0.00 | metal
MS9 | [0110100] | 0.08088 | 5.4703 (5.4248) | 0.00 | metal
MS10 | [1100001] | 0.09682 | 5.4151 (5.4954) | 0.00 | metal
MS11 | [1100010],[1010001] | 0.11800 | 5.4022 (5.4818) | 0.00 | metal

It was already shown [10] that when the symmetries of wavefunctions were not
included in the solutions of the KS equations, all metastable states turn into
insulators with different gaps. In the case of OMC-$C^{7}_{3}$ method, we
solved the KS equations without any symmetries for the wavefunctions other
than the time reversal symmetry, and most of metallic metastable states turned
into insulators. These calculations were done for cubic structure with
experimental lattice constant of 5.47Å. The results are shown in Table 2.

Table 2: Band gaps of metastable states for OMC-$C^{7}_{3}$ method when the symmetry of the wavefunctions were not included in the solutions of the KS equations. The states are sorted by the initial configurations of occupation matrices. State | initial occupations |  $E_{gap}$ (eV)
---|---|---
GS* | [1110000] | 2.90
MS*1 | [1101000] | 2.19
MS*2 | [1100100] | 2.51
MS*3 | [1100010] | 2.83
MS*4 | [1100001] | 1.89
MS*5 | [1011000] | 2.58
MS*6 | [1010100] | 2.51
MS*7 | [1010010] | 1.89
MS*8 | [1010001] | 2.79
MS*9 | [1001100] | 2.15
MS*10 | [1001010] | 2.36
MS*11 | [1001001] | 2.36
MS*12 | [1000110] | 2.38
MS*13 | [1000101] | 2.36
MS*14 | [1000011] | 2.21
MS*15 | [0111000] | not conv.
MS*16 | [0110100] | not conv.
MS*17 | [0110010] | 2.84
MS*18 | [0110001] | 2.84
MS*19 | [0101100] | 2.57
MS*20 | [0101010] | 2.57
MS*21 | [0101001] | 2.21
MS*22 | [0100110] | 2.82
MS*23 | [0100101] | not conv.
MS*24 | [0100011] | metal
MS*25 | [0011100] | 2.57
MS*26 | [0011010] | 2.21
MS*27 | [0011001] | 2.57
MS*28 | [0010110] | not conv.
MS*29 | [0010101] | 2.86
MS*30 | [0010011] | metal
MS*31 | [0001110] | 2.35
MS*32 | [0001101] | 2.35
MS*33 | [0001011] | metal
MS*34 | [0000111] | not conv.

The states tabulated in Table 2 are named according to the configuration order
and not the energy order.

####  III.1.2 $C_{2}^{7}$ occupations

To test the validity of arguments [10] which led to $C_{2}^{7}$
configurations, we have performed the calculations and tabulated the results
in Table 3. Comparison of Table 1 and Table 3 reveals that the GS and first
two metastable states are the same in $C_{3}^{7}$ and $C_{2}^{7}$ schemes.
However, the latter scheme predicts two more metastable states with gap. So,
$C_{2}^{7}$ is not the only scheme to find the GS. In addition, we see that
the predicted lattice constant for the GS, 5.4656(5.4787) $\AA$ is in
excellent agreement with experimental value, 5.47 $\AA$. However, the
predicted KS band gap, 2.91 eV, is about 0.7 eV larger than the experimental
gap.

Table 3: The same as in Table 1 but for the OMC-$C^{7}_{2}$ method.  State | initial occupations | $\Delta E$ |  $a$ ($c$) | Tot. mag. | $E_{gap}$
---|---|---|---|---|---
GS | [110000] | 0.00000 | 5.4656 (5.4787) | 0.00 | 2.91
MS1 | [1000001],[0000011] | 0.00464 | 5.4888 (5.4327) | 0.00 | 2.22
MS2 | [0001100] | 0.00838 | 5.4754 (5.4523) | 0.00 | 2.15
MS3 | [0010010] | 0.02426 | 5.5068 (5.3983) | 0.00 | metal
MS4 | [0001010],[0001001] | 0.05790 | 5.4354 (5.4628) | 0.00 | metal
MS5 | [0000110],[0000101] | 0.06203 | 5.4362 (5.4694) | 0.00 | metal
MS6 | [0101000],[0011000] | 0.06206 | 5.4444 (5.4882) | 0.00 | metal
MS7 | [1000100] | 0.06370 | 5.5209 (5.5176) | 0.00 | 1.00
MS8 | [1001000] | 0.06608 | 5.5265 (5.5020) | 0.00 | 0.82
MS9 | [0100100],[0010100] | 0.06865 | 5.4588 (5.4571) | 0.00 | metal
MS10 | [0100001] | 0.09682 | 5.4151 (5.4954) | 0.00 | metal
MS11 | [0100010],[0010001] | 0.11800 | 5.4022 (5.4818) | 0.00 | metal
MS12 | [1100000],[1010000] | 0.13873 | 5.4992 (5.5028) | 0.00 | metal

###  III.2 SMC results

It was shown [21] that for the all-equivalent-O scheme, one obtains seven
different classes of energetic and structural properties. Also, the calculated
total magnetizations
$M_{tot}=\int_{cell}{\left(n_{\uparrow}-n_{\downarrow}\right)d^{3}r}$ had
shown that the GS as well as the six meta-stable states had zero total
magnetizations. The important result in the SMC method is that, in contrast to
that of OMC method, for the GS the spin-up and spin-down electrons of O atoms
are not symmetrically distributed. That is, the GS has a value $\zeta\neq 0$.
The situation is schematically shown in Fig. 3.

Figure 3: Schematic plot of local minima and their corresponding starting
magnetizations. The depths of the minima (in arbitrary units) are consistent
with their energy values. As is seen, the GS and GS’ cover the largest
interval, but away from the zero starting magnetization.

In the two-inequivalent-O scheme, we have optimized the structures for 441
different combinations of $\zeta_{1}$ and $\zeta_{2}$ values and obtained 16
local minima for the energies. The energy results are shown in Fig. 4.

Figure 4: Local-minimum states in two-inequivalent-O scheme. It is seen that
for the GS, the values of $\zeta_{1}$ and $\zeta_{2}$ are not simultaneously
zero and most of $\zeta_{1}$ and $\zeta_{2}$ combinations lead to the GS than
the other 15 metastable states. Also invariance under
$(\zeta_{1},\zeta_{2})\rightarrow(-\zeta_{1},-\zeta_{2})$ transformation is
clearly seen.

As is seen from Fig. 4, there are 16 local-minimum states and most of
$\zeta_{1}$ and $\zeta_{2}$ combinations lead to the GS. This figure also
shows that the properties are invariant under the transformation
$(\zeta_{1},\zeta_{2})\rightarrow(-\zeta_{1},-\zeta_{2})$.

Here, we have also performed SCF calculation with no symmetry (other than the
time-reversal) for the wavefunctions, and did not find any significant
differences in the number of ”insulator” metastable states, and therefore do
not report the results.

Table 4: GS and meta-stable (MS) states’ properties in the SMC method with two-inequivalent-O atoms. The energies are in Ry/(formula unit) and are compared to the GS. Equilibrium lattice constants are in $\AA$, total magnetizations are in Bohr-magneton/(formula unit), and energy gap is in eV. Because of $(\zeta_{1},\zeta_{2})\rightarrow(-\zeta_{1},-\zeta_{2})$ symmetry in properties, they are not brought here.  State | {($\zeta_{1}$ , $\zeta_{2}$)} | $\Delta E$ |  $a$ ($c$) | Tot. mag. | $E_{gap}$
---|---|---|---|---|---
GS | (-1.0, [-0.7,+0.2]), (-0.9, [-0.8,+0.3]), (-0.8, [-0.9,+0.3]), | 0.00000 | 5.4765 (5.4566) | 0.00 | 2.18
| (-0.7, [-1.0,+0.1]), (-0.6, [-1.0,0.0]), (-0.5, [-1.0,-0.1]), |  |  |  |
| (-0.4, [-1.0,-0.2]), (-0.3, [-1.0,-0.3]), (-0.3, [+0.8,+0.9]), |  |  |  |
| (-0.2, [-1.0,-0.4]), (-0.2, [+0.8,+1.0]), (-0.1, [-1.0,-0.5]), |  |  |  |
| (-0.1, [+0.7,+1.0]), (0.0, [-1.0,-0.6]), (0.0, [+0.6,+1.0]) |  |  |  |
MS1 | (-0.7, [+0.4,+0.6]), (-0.6, [+0.3,+0.7]), (-0.5, [+0.2,+0.7]), | 0.00240 | 5.4888 (5.4327) | 0.00 | 2.22
| (-0.4, [+0.2,+0.7]), (-0.3, [+0.1,+0.6]), (-0.2, [0.0,+0.5]), |  |  |  |
| (-0.1, [-0.1,+0.4]), (0.0, [-0.2,+0.2]) |  |  |  |
MS2 | (-1.0, [+0.9,+1.0]), (-0.9, [+0.9,+1.0]), (-0.8,+1.0) | 0.00614 | 5.4755 (5.4526) | 0.00 | 2.15
MS3 | (-0.8,+0.4), (-0.7, [+0.2,+0.3]), (-0.6, [+0.1,+0.2]), | 0.01218 | 5.4981 (5.4150) | 0.00 | 1.90
| (-0.5, [0.0,+0.1]), (-0.4, [-0.1,+0.1]), (-0.4,+0.8), |  |  |  |
| (-0.3, [-0.2,0.0]), (-0.3, +0.7), (-0.2, [-0.3,-0.1]), |  |  |  |
| (-0.2, [+0.6,+0.7]), (-0.1, [-0.4,-0.2]), (-0.1, [+0.5,+0.6]), |  |  |  |
| (0.0, [-0.5,-0.3]), (0.0, [+0.3,+0.5]) |  |  |  |
MS4 | (-1.0, [-0.9,-0.8]), (-0.9, -1.0), (-0.8, -1.0) | 0.02848 | 5.4504 (5.4746) | 0.00 | metal
MS5 | (-0.8, +0.5), (-0.5, +0.8) | 0.029225 | 5.4621 (5.4485) | 0.00 | metal
MS6 | (-1.0, [+0.6,+0.7]), (-0.9, +0.8), (-0.7, +1.0), | 0.031185 | 5.4562 (5.4578) | 0.00 | metal
| (-0.6, +1.0) |  |  |  |
MS7 | (-1.0, +0.3) | 0.031215 | 5.4719 (5.4516) | 0.00 | metal
MS8 | (-0.9, +0.4), (-0.4, +0.9) | 0.031775 | 5.4533 (5.4720) | 0.00 | metal
MS9 | (-0.9, -0.9) | 0.032190 | 5.4552 (5.4590) | 0.00 | metal
MS10 | (-0.3, +1.0) | 0.038175 | 5.4705 (5.4507) | 0.00 | metal
MS11 | (-1.0, +0.8) | 0.042295 | 5.4559 (5.4330) | 0.00 | metal
MS12 | (-0.9, [+0.5,+0.7]), (-0.8, [+0.6,+0.8]), (-0.7, [+0.7,+0.9]), | 0.055675 | 5.4354 (5.4628) | 0.00 | metal
| (-0.6, [+0.8,+0.9]), (-0.5, +0.9) |  |  |  |
MS13 | (-1.0, [+0.4,+0.5]), (-0.5, +1.0), (-0.4, +1.0) | 0.058130 | 5.4257 (5.4870) | 0.00 | metal
MS14 | (-0.8, +0.9) | 0.064050 | 5.4390 (5.4467) | 0.00 | metal
MS15 | (-1.0, -1.0) | 0.066290 | 5.4315 (5.5120) | 0.00 | metal

Finally, we address the most important differences of the ground state
properties in the two schemes of OMC and SMC.

(a)

(b)

Figure 5: Density of states (DOS) for spin-up (purple color) and spin-down
(green color) for (a)- OMC and (b)- SMC methods. In OMC it is symmetric but in
SMC the symmetry is broken.

Firstly, the GS energy obtained in SMC is slightly higher than that of OMC by
0.0022 Ry/(formula unit). Secondly, the KS gap in SMC-GS, 2.18 eV, is very
close to the experimental value compared to that of OMC-GS, 2.91 eV; and
thirdly, as is seen from Fig. 5, in the GS of OMC scheme the up-spin and down-
spin electrons are symmetrically distributed over the states while in the SMC
scheme even though the total magnetization vanishes but the symmetry between
up and down spins is broken.

##  IV Conclusions

To overcome the incorrect prediction of metallic behavior for UO2 by ordinary
DFT method, one may use a Hubbard correction for the energy contributions of
localized orbitals, which is called the DFT+U method. However, the application
of DFT+U to determine the GS properties is somewhat tricky in that one should
take care not to stuck in higher-energy metastable states which have very
different structural and electronic properties. The occupation-matrix-control
(OMC) method was one way to search for the global minimum among lots of local
minimum states. In the OMC method, people constrain the starting magnetization
of O atoms to be zero and use different possible initial occupation matrices
for the localized orbital. However, in the SMC method we release the constrain
used in the OMC method and let the starting magnetizations of O atoms vary
between -1.0 and +1.0. It turned out that in this way one reproduces the
metastable states. It was shown that in the SMC one obtains more metastable
states than that of OMC. Comparing the GS properties in the two method showed
very similar lattice constants and total magnetizations. However, the band gap
in the SMC was in very good agreement with experiment than that of OMC.

## Acknowledgement

This work is part of research program in School of Physics and Accelerators,
NSTRI, AEOI.

## Data availability

The raw or processed data required to reproduce these results can be shared
with anybody interested upon sending an email to M. Payami.

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