The following is a problem from a text on Critical Point Theory I am reading. Below is the beginning of an attempt, but I got stuck. Any hints will be the most appreciated. Thanks in advance and kind regards.
Use the Ekeland Variational Principle to solve the following sublinear problem: $$ (P) \quad \begin{cases} -\Delta_p u + |u|^{p - 2}u = h(x)|u|^{q - 2}u \quad \text{ in }\Bbb{R}^N \\ u \in W^{1, p}(\Bbb{R}^N) \end{cases} $$ where $\Delta_p$ is the $p$-laplace operator, $N \geq 3$, $2 \leq p < N$, $p - 1 < q < p$, $h \in L^{\frac{p^*}{p^* - q}}(\Bbb{R}^N) \cap L^\infty (\Bbb{R}^N)$, $h \geq 0$ and $h \neq 0$.
Weak solutions to the problem $(P)$ are critical points of the functional \begin{align*} I(u) & = \frac1p \int_{\Bbb{R}^N} |\nabla u|^p \ dx+ \frac1p \int_{\Bbb{R}^N} |u|^p \ dx - \frac1q \int_{\Bbb{R}^N} h(x) |u|^q \ dx \\ & = \frac1p ||u||^p - \frac1q \int_{\Bbb{R}^N} h(x)|u|^q \ dx, \quad u \in W^{1, p} (\Bbb{R}^N) \end{align*} which is of class $C^1$, with $$ I'(u)v = \int_{\Bbb{R}^N} |\nabla u|^{p - 2} \nabla u \cdot \nabla v \ dx + \int_{\Bbb{R}^N} |u|^{p - 2}uv \ dx - \int_{\Bbb{R}^N} h(x) |u|^{q - 2}uv \ dx, $$ for $u, v \in W^{1, p}(\Bbb{R}^N)$. The functional $I$ is also weakly lower semicontinuous and coercive, and hence bounded from below. Therefore, if $c = \inf_{W^{1, p}(\Bbb{R}^N)}I$, by the Ekeland Variational Principle there exists a Palais-Smale sequence at the level $c$.
Let $(u_n)$ be a $(PS)_c$ sequence for $I$. Then $(u_n)$ is bounded, since $I$ is coercive. Since $W^{1, p}(\Bbb{R}^N)$ is reflexive, there exists $u \in W^{1, p} (\Bbb{R}^N)$ such that $u_n \rightharpoonup u$.
Let $\phi \in C_c^\infty (\Bbb{R}^N)$ and let $\Omega = \text{supp} \phi$. Then $$ u_n|_\Omega \rightharpoonup u|_\Omega \quad \text{ in } W^{1, p}(\Omega) $$ and therefore, by the compact Sobolev embeddings, $$ u_n|_\Omega \to u|_\Omega \quad \text{ in } L^s(\Omega) $$ for $s \in [1, p^*)$, up to a subsequence. It is can be shown that $$ \int_{\Bbb{R}^N} h(x) |u_n|^{q - 2} u_n \phi \ dx \to \int_{\Bbb{R}^N} h(x) |u|^{q - 2} u \phi \ dx, $$ as well as that $$ \int_{\Bbb{R}^N} |u_n|^{p - 2} u_n \phi \ dx \to \int_{\Bbb{R}^N} |u|^{p - 2} u \phi \ dx, $$ which holds for all $\phi \in C_c^\infty(\Bbb{R}^N)$. It remains to show that \begin{align*} \int_{\Bbb{R}^N} |\nabla u_n|^{p - 2} \nabla u_n \cdot \nabla \phi \ dx \to \int_{\Bbb{R}^N} |\nabla u|^{p - 2} \nabla u \cdot \nabla \phi \ dx \end{align*}
I found a way to prove the desired convergence. It is as follows. I would appreciate any critique and comments.
Choose $\phi \in C_c^\infty(\Bbb{R}^N)$. Let $\psi \in C_c^\infty(\Bbb{R}^N)$ be such that $0 \leq \psi \leq 1$ and $$ \psi(x) = \begin{cases} 1, \quad x \in B_1(0) \\ 0, \quad x \in \Bbb{R}^N \setminus B_2(0) \end{cases} $$ For each $\rho > 0$, let $$ \psi_\rho = \psi \left(\frac x\rho \right). $$ Then $$ \psi_\rho = \begin{cases} 1, \quad x \in B_\rho(0) \\ 0, \quad x \in \Bbb{R}^N \setminus B_{2\rho}(0) \end{cases}. $$ Defining $$ P_n(x) = (|\nabla u_n|^{p-2} \nabla u_n - |\nabla u|^{p - 2} \nabla u) \cdot (\nabla u_n - \nabla u) $$ we have that \begin{align*} 0 & \leq C_p \int_{B\rho(0)} |\nabla u_n - \nabla u|^p \ dx \\ & \leq \int_{B\rho(0)} P_n(x) \ dx \\ & \leq \int_{B\rho(0)} P_n(x) \psi_\rho(x) \ dx \\ & \leq \int_{\Bbb{R}^N} P_n(x) \psi_\rho(x) \ dx. \end{align*} Therefore \begin{align*} 0 & \leq C_p \int_{B\rho(0)} |\nabla u_n - \nabla u|^p \ dx \\ & \leq \int_{\Bbb{R}^N}|\nabla u_n|^p \psi_\rho \ dx - \int_{\Bbb{R}^N} |\nabla u_n|^{p - 2} \nabla u_n \cdot \nabla u \psi_\rho \ dx - \int_{\Bbb{R}^N} |\nabla u|^{p - 2} \nabla u \cdot \nabla u_n \psi_\rho \ dx \\ & \quad + \int_{\Bbb{R}^N} |\nabla u|^p \psi_\rho \ dx \\ & = J_1(u_n) - J_2(u_n) + J_3(u_n) + J_4(u_n) + J_5(u_n), \end{align*} where $$ J_1(u_n) = \int_{\Bbb{R}^N} |\nabla u_n|^p \psi_\rho \ dx + \int_{\Bbb{R}^N} |u_n|^p \psi_\rho - \int_{\Bbb{R}^N} h(x) |u_n|^q \psi_\rho \ dx, $$ \begin{align*} J_2(u_n) = & \int_{\Bbb{R}^N} |\nabla u_n|^{p - 2} \nabla u_n \cdot \nabla u \psi_\rho \ dx + \int_{\Bbb{R}^N} |u_n|^{p - 2} u_n u \psi_\rho \ dx \\ & - \int_{\Bbb{R}^N} |u_n|^{q - 2} u_n u \psi_\rho \ dx, \end{align*} $$ J_3(u_n) = - \int_{\Bbb{R}^N} |\nabla u|^{p - 2} \nabla u \cdot \nabla u_n \psi_\rho \ dx + \int_{\Bbb{R}^N} |\nabla u|^p \psi_\rho \ dx, $$ $$ J_4(u_n) = \int_{\Bbb{R}^N} |u_n|^{p - 2} u_n u \psi_\rho \ dx - \int_{\Bbb{R}^N} |u_n|^p \psi_\rho \ dx $$ and $$ J_5(u_n) = \int_{\Bbb{R}^N} |u_n|^q \psi_\rho \ dx - \int_{\Bbb{R}^N} |u_n|^{q - 2}u_nu \psi_\rho \ dx. $$
We begin by noting that $$ J_1(u_n) = I'(u_n)(u_n \psi_\rho) - \int_{\Bbb{R}^N} |\nabla u_n|^{p - 2} \nabla u_n \cdot \nabla \psi_\rho u_n \ dx $$ and also that \begin{align*} ||u_n \psi_\rho||^p & = \int_{\Bbb{R}^N} |\nabla u_n \psi_\rho|^p \ dx + \int_{\Bbb{R}^N}|u_n \psi_\rho|^p \ dx \\ & \leq C||u_n||^p \\ & \leq C_1 \end{align*} for some $C_1 > 0$, since the sequence $(u_n)$ is bounded. But then, since $I'(u_n) \to 0$, $$ J_1(u_n) = o_n(1) - \int_{\Bbb{R}^N} |\nabla u_n|^{p - 2} \nabla u_n \cdot \nabla \psi_\rho u_n \ dx. $$ On the other hand, note that \begin{align*} \left|\int_{\Bbb{R}^N} |\nabla u_n|^{p-2} \nabla u_n \cdot \nabla \psi_\rho u_n \ dx \right| & \leq \int_{\Bbb{R}^N} |\nabla u_n|^{p - 1} |\nabla \psi_\rho| |u_n| \ dx \\ & \leq \left(\int_{\Bbb{R}^N}|\nabla u_n|^p \ dx\right)^{\frac{p-1}{p}} \left(\int_{\Bbb{R}^N} |\nabla \psi_\rho|^p |u_n|^p \ dx \right)^{\frac1p} \\ & \leq C_1 \left(\int_{B_{2\rho}(0) \setminus B_\rho(0)} |\nabla \psi_\rho|^p |u_n|^p \ dx \right)^{\frac1p} \end{align*} where the first inequality follows from Cauchy-Schwarz, the second from Hölder's Inequality with exponents $p/(p - 1)$ and $p$, and the third by the boundedness of $(u_n)$. Now, note that $u_n \to u$ in $L^p(B_{2\rho}(0) \setminus B_\rho(0))$. Then, applying Vainberg's Theorem and the Dominated Convergence Theorem in sequence yields $$ \limsup_{n \to \infty} \left|\int_{\Bbb{R}^N} |\nabla u_n|^{p-2} \nabla u_n \cdot \nabla \psi_\rho u_n \ dx \right| \leq C_1 \left(\int_{B_{2\rho}(0) \setminus B_\rho(0)} |\nabla \psi_\rho|^p |u|^p \ dx \right)^{\frac1p}. $$ From Hölder's Inequality with exponents $N/(N - p)$ and $N/p$ it follows that \begin{align*} \limsup_{n \to \infty} & \left|\int_{\Bbb{R}^N} |\nabla u_n|^{p-2} \nabla u_n \cdot \nabla \psi_\rho u_n \ dx \right| \\ & \leq C_1 \left[ \left(\int_{B_{2\rho}(0) \setminus B_\rho(0)} |u|^{p^*} \ dx \right)^{\frac{N - p}{p}} \left(\int_{B_{2\rho}(0) \setminus B_\rho(0)} |\nabla \psi_\rho|^N \right)^{\frac Np} \right]^{\frac1p} \\ & \leq C_1 \left[ \left(\int_{B_{2\rho}(0) \setminus B_\rho(0)} |u|^{p^*} \ dx \right)^{\frac{N - p}{p}} \left(\int_{\Bbb{R}^N} |\nabla \psi|^N \right)^{\frac Np} \right]^{\frac1p}. \end{align*} Then, by the Dominated Convergence Theorem, $$ \lim_{\rho \to 0} \limsup_{n \to \infty} \left|\int_{\Bbb{R}^N} |\nabla u_n|^{p-2} \nabla u_n \cdot \nabla \psi_\rho u_n \ dx \right| = 0 $$ and therefore $$ J_1(u_n) = o_n(1) + o_\rho(1). $$ By an analogous argument, $$ J_2(u_n) = o_n(1) + o_\rho(1). $$ By the weak convergence, $$ J_3(u_n) = o_n(1). $$ Since $u_n \to u$ in $L^s_{\text{loc}}$ and $\psi_\rho$ has compact support, the Dominated Convergence Theorem yields $$ J_4(u_n) = o_n(1) $$ and $$ J_5(u_n) = o_n(1). $$ It therefore follows that $$ \frac{\partial u_n}{x_i} \to \frac{\partial u}{x_i} \quad \text{ in } L^p_{\text{loc}} (\Bbb{R}^N) $$ for all $i \in \{1, \ldots, N\}$. In particular, $$ \left. \frac{\partial u_n}{x_i}\right|_{B_R(0)} \to \left. \frac{\partial u}{x_i}\right|_{B_R(0)} \quad \text{ in } L^p(B_R(0)) \ \forall R > 0. $$ By Vainberg's Theorem, there exists a subsequence $(u_{1n}) \subset (u_n)$ such that $$ \frac{\partial u_{1n}}{x_i} (x) \to \frac{\partial u}{x_i}(x) \quad \text{ a.e. in } B_1(0). $$ Now, by the compact Sobolev embedding on the sequence $(u_{1n})$ there exists a subsequence $(u_{2n})$ such that $$ \frac{\partial u_{2n}}{x_i} (x) \to \frac{\partial u}{x_i}(x) \quad \text{ a.e. in } B_2(0). $$ Proceeding in an analogous manner, for every $k \in \Bbb{N}$ there exists $(u_{kn}) \subset (u_n)$ such that $$ \frac{\partial u_{kn}}{x_i} (x) \to \frac{\partial u}{x_i}(x) \quad \text{ a.e. in } B_k(0). $$ We claim that $(u_{jj})$ is such that $$ \frac{\partial u_{jj}}{\partial x_i}(x) \to \frac{\partial u}{\partial x_i}(x) \quad \text{ a.e in } \Bbb{R}^N. $$ Let $$ S_k = \left\{x \in B_k(0) \ : \ \frac{\partial u_{kn}}{x_i} (x) \not\to \frac{\partial u}{x_i}(x) \right\} $$ and $S = \cap_k S_k$. It is clear that $|S| = 0$, since it is a countable union of sets of measure $0$. Let $x \in \Bbb{R}^N \setminus S$ and $j_0 \in \Bbb{N}$ such that $x \in B_{j_0}(0)$. Then $x \in B_j(0)$ for all $j \geq j_0$. Moreover, $$ \frac{\partial u_{j_0n}}{\partial x_i}(x) \to \frac{\partial u}{\partial x_i}(x) \quad \text{ a.e in } B_{j_0}(0). $$ Since $(u_{jj})$ is a subsequence of $(u_{j_0n})$, the claim follows. Therefore it holds that $$ |\nabla u_n|^{p - 2}\nabla u_n \to |\nabla u|^{p - 2} \nabla u \quad \text{ a.e. in } \Bbb{R}^N. $$ Moreover, the sequence $(|\nabla u_n|^{p - 2}\nabla u_n)$ is bounded in $L^{\frac{p}{p - 1}}$. Hence, by the Brezis-Lieb Lemma, $$ \int_{\Bbb{R}^N} |\nabla u_n|^{p - 2} \nabla u_n \cdot \nabla \phi \ dx \to \int_{\Bbb{R}^N} |\nabla u|^{p - 2} \nabla u \cdot \nabla \phi \ dx. $$