I am studying the Ekeland Variational Principle, and doing some exercises on the subject, but I lack confidence. I would be very thankful if someone could read my solution and verify its correctness or point out any mistakes.
Thanks in advance and kind regards.
Problem Use the Ekeland Variational Principle to solve the following sublinear problem: $$ (P)\quad \begin{cases} - \Delta_pu = |u|^{q - 2}u \quad \text{ in } \Omega, \\ u = 0 \quad \text{ on } \partial \Omega \end{cases} $$ where $\Omega$ is a bounded domain of $\Bbb{R}^N$, $N \geq 3$, $2 \leq p \leq N$ and $p - 1 \leq q \leq p$. $\Delta_p$ denotes the $p$-laplacian operator.
Solution
A weak solution to problem $(P)$ is a critical point of the functional $$ I(u) = \frac1p \int_\Omega |\nabla u|^p \ dx - \frac1q \int_\Omega |u|^q \ dx = \frac1p ||u||^p - \frac1q ||u||_q^q, \quad u \in W_0^{1, p} (\Omega), $$ which is of class $C^1$ with $$ I'(u)v = \int_\Omega |\nabla u|^{p - 2} \nabla u \cdot \nabla v \ dx - \int_\Omega |u|^{q - 2}uv \ dx, \quad v \in W_0^{1, p} (\Omega). $$
We know that $I$ is weakly lower semicontinuous and coercive. Then it is bounded from below. Let $c = \inf_{W_0^{1, p} (\Omega)}I$. By the Ekeland Variational Principle, there exists a Palais-Smale sequence at the level $c$. If we show that $I$ satisfies $(PS)_c$, we are done.
Let $(u_n) \subset W_0^{1, p} (\Omega)$ be a Palais-Smale sequence at the level $c$, that is, $$ I(u_n) \to c, \quad I'(u_n) \to 0. $$ We claim that $(u_n)$ has a convergent subsequence. First, note that $(u_n)$ is bounded. Since $W_0^{1, p} (\Omega)$ is reflexive, there is some $u \in W_0^{1, p}(\Omega)$ such that $u_n \rightharpoonup u$ in $W_0^{1, p} (\Omega)$. Then, by the compact Sobolev embeddings, we have that $$ u_n \to u \quad \text{ in } L^s(\Omega), $$ for $s \in [1, p^*)$. In particular, this convergence holds for $s = q$.
Now, we want to show that $$ ||u_n - u||^p = \int_\Omega |\nabla(u_n - u)|^p \ dx \to 0. $$ But note that $$ \frac1p ||u_n - u||^p = I'(u_n - u)(u_n - u) + \frac1q ||u_n - u||_q^q \to 0, $$ since $I'(u_n) \to 0$ since $(u_n)$ is a Palais-Smale sequence, $I'(u)(u_n - u) \to 0$ by the weak convergence and $u_n - u \to 0$ in $L^q(\Omega)$ by the compact Sobolev embedding. Hence $u_n \to u$ in $W_0^{1, p}(\Omega)$ and the claim is proved.
The limit $u$ is a critical point of $I$, hence a weak solution to problem $(P)$. It only remains to show that $u$ is not the trivial solution. Let $\psi \in C_c^\infty (\Omega)$ such that $\psi > 0$. Then for $t > 0$ we have that $$ I(t \psi) = \frac{t^p}{p} ||\psi||^p - \frac{t^q}{q} ||\psi||_q^q < 0 $$ if $t$ is sufficiently small. Hence $I(u) < 0 = I(0)$, so $u$ is not the trivial solution.