$w=\mathcal L_K^{-1}(\lambda u+f(x,u))$ and using the definitions of $\mathcal L_K$ and $\mathcal J_\lambda$, for any test function $\varphi\in X_0$ we get that $$\begin{aligned}
\langle w, \varphi \rangle_{X_0} & = \lambda \int_\Omega u(x)\varphi(x)\,dx +\int_\Omega f(x,u(x))\varphi(x)\,dx\\
& = - \langle \mathcal J_\lambda'(u), \varphi\rangle+\langle u, \varphi \rangle_{X_0}\\
& = -\langle \nabla \mathcal J_\lambda(u),v\rangle_{X_0}+\langle u, \varphi \rangle_{X_0}\,,
\end{aligned}$$ that is $$w=-\nabla \mathcal J_\lambda(u)+u\,,$$ which gives the assertion.

Moreover, we note that $\mathcal L_K^{-1}$ is a compact operator for all $\nu\in [1,2^*)$. Indeed, if $\{g_n\}_{n\in {\mathbb N}}$ is bounded in $L^{\nu'}(\Omega)$, standard calculations imply that $\left\{\mathcal L_K^{-1}g_n\right\}_{n\in {\mathbb N}}$ is relatively compact in $X_0$.

For further use, we also note that, again using the definition of $\mathcal L_K^{-1}$, $$\label{scarica}
\langle u, \mathcal L_K^{-1}v\rangle_{X_0}=\int_\Omega u(x)v(x)\,dx$$ for every $u,v\in X_0$.

Compactness condition {#sec:compactness}
=====================

In this section we check the validity of the *Palais–Smale condition* for functional $\mathcal J_\lambda$ at any level, that is we prove that for each $c\in {\mathbb R}$ any Palais–Smale sequence for $\mathcal J_\lambda$ at level $c$ admits a subsequence which is strongly convergent in $X_0$. As usual, we say that $\{u_j\}_{j\in {\mathbb N}}\subset X_0$ is a *Palais–Smale sequence* for $\mathcal J_\lambda$ at level $c\in {\mathbb R}$ if $$\label{Jc0}
\mathcal J_\lambda(u_j)\to c$$ and 