is a solution of . Moreover, from Dancer and Yan [@DY], we know that $w$ is also non-degenerate, in the sense that the kernel of the operator $Lv:=-\Delta v- pw_+^{p-1} v,~~v\in
D^{1,2}(\mathbb{R}^2)$ is spanned by $\bigl\{\frac{\partial
w}{\partial x_1}, \frac{\partial w}{\partial x_2}\bigr\}$.

Recall that $Z=(z_1,\cdots,z_m)$, and $z_j\in{\Omega}$ satisfies

$$\label{3.3}
d(z_j,\partial{\Omega})\ge \varrho>0,\quad |z_i-z_j|\ge \varrho^{\bar
L},\qquad i\ne j,$$

where $\varrho>0$ is a fixed small constant, and $\bar L>0$ is a large constant.

Let $P_{{\delta},Z,j}$ be the function defined in . Set

$$F_{{\delta},Z}=\left\{u: u\in L^p({\Omega}),\; \int_{\Omega}\frac{\partial
P_{{\delta},Z,j}}{\partial z_{j,h}}u =0, j=1,\cdots,m,\;h=1,2 \right\},$$ and

$$E_{{\delta},Z}=\left\{u:\;u\in W^{2,p}({\Omega})\cap H_0^1({\Omega}), \int_{\Omega}\Delta\left( \frac{\partial
P_{{\delta},Z,j}}{\partial z_{j,h}}\right)u =0, j=1,\cdots,m,\; h=1,2
\right\}.$$

For any $u\in L^p({\Omega})$, define $Q_{\delta}u$ as follows: $$Q_{\delta}u= u-\sum_{j=1}^m \sum_{h=1}^2
b_{j,h}\left(-{\delta}^2\Delta\Bigl(\frac{\partial P_{{\delta},Z,j}}{\partial
z_{j,h}}\Bigr)
 \right),$$ where the constants $b_{j,h}$, $j=1,\cdots,m$, $h=1, 2$, satisfy

$$\label{3.4}
\sum_{j=1}^m \sum_{h=1}^2 b_{j,h}\left(-{\delta}^2\int_{{\Omega}}\Delta\Bigl(
\frac{\partial P_{{\delta},Z,j}}{\partial z_{j, h}}\Bigr) \frac{\partial
P_{{\delta},Z,i}}{\partial z_{i,\bar h}}\right)=\int_{{\Omega}} u
\frac{\partial P_{{\delta},Z,i}}{\partial z_{i,\bar h}}.$$

Since $\int_{\Omega}\frac{\partial
P_{{\delta},Z,j}}{\partial z_{j,h}} Q_{\delta}u=0$, the operator $Q_{\delta}$ can be regarded as a projection from $L^p(\Omega)$ to $F_{{\delta},Z}$. In order to show that we can solve to obtain $b_{j,h}$, we just need the following estimate ( by and ):

$$\label{3.5}
\begin{split}
&-{\delta}^2\int_{{\Omega}}\Delta\Bigl( \frac{\partial P_{{\delta},Z,j}}{\partial
z_{j, h}}\Bigr)
\frac{\partial P_{{\delta},Z,i}}{\partial z_{i,\bar h}}\\
=&
p\int_{{\Omega}}\bigl(W_{{\delta},z_j,a_{{\delta},j}}-a_{{\delta},j}\bigr)_+^{p-1}\left(\frac{\partial
W_{{\delta},z_{j},a_{{\delta},j}}}{\partial z_{j, h}}-\frac{\partial
a_{{\delta},j}}{\partial z_{j,h}}\right) \frac{\partial
P_{{\delta},Z,i}}{\partial z_{i,\bar h}}
