The following is in Appendix B of Struwe's Variational Methods
Let $u$ be a solution of $-\Delta u = g(x, u(x))$ in a domain $\Omega \subset \mathbb R^N$, $N \geq 3$, where $g$ is a Carathéodory function with subcritical superlinear growth.
Theorem:
Let $\Omega \subset \mathbb R^N$ be a smooth open set and let $g: \Omega \times \mathbb R \to \mathbb R$ be a Carathéodory function such that
$$
|g(x, u(x))| \leq a(x)(1 + |u(x)|) \quad \text{ a.e. in } \Omega
$$
for some $0 \leq a \in L_{loc}^{N/2}(\Omega)$. Let $u \in H^1_{loc}(\Omega)$ be a weak solution to $-\Delta u = g(x, u)$. Then $u \in L^q_{loc}(\Omega)$ for all $1 < q < \infty$. If $u \in H_0^1(\Omega)$ and $a \in L^{N/2}(\Omega)$, then $u \in L^q(\Omega)$ for all $1 < q < \infty$.
The proof goes as follows:
Take $\eta \in C_c^\infty(\Omega)$, $s \geq 0$ and $L \geq 0$ and let
$$
\varphi = u \min \{|u|^{2s}, L^2\} \eta^2 \in H_0^1(\Omega)
$$
Testing the equation against $\varphi$ yields
$$
\int_\Omega |\nabla u|^2 \min\{|u|^{2s}, L^2\} \eta^2 \ dx + \frac s2 \int_{\{|u|^s\leq L \}} |\nabla(|u|^2)|^2 |u|^{2s – 2} \eta ^2 \ dx \\
\leq -2 \int_\Omega \nabla u u \min \{|u|^{2s}, L^2\} \nabla \eta \eta \ dx + \int_\Omega a(1 + 2|u|^2)\min \{|u|^{2s}, L^2\}\eta^2 \ dx \\
(*) \quad {\leq} \frac 12 \int_\Omega |\nabla u|^2 \min\{|u|^{2s}, L^2\}\eta^2 \ dx + c \int_\Omega |u|^2 \min\{|u|^{2s}, L^2\} |\nabla \eta|^2 \ dx \\
\quad + 3 \int_\Omega a|u|^2 \min\{|u|^{2s}, L^2\} \eta^2 \ dx + \int_\Omega |a|\eta^2 \ dx
$$
Why does $(*)$ hold?
Thanks in advance and kind regards.
Best Answer
I got this with the hint by MaoWao and the help of David Stolnicki.
Recall Young's Inequality with an epsilon: $$ ab \leq \frac 1{2\varepsilon} a^2 + \frac \varepsilon2 b^2. $$ We apply this inequality to the first term, which gives \begin{align*} - 2 \int_\Omega \nabla u u \min\{|u|^{2s}, L^2\} \nabla \eta \eta \ dx \leq & \frac 12 \int_\Omega |\nabla u|^2 \min \{|u|^{2s}, L^2\} \eta^2 \ dx \\ & + c \int_\Omega u^2 \min\{|u|^{2s}, L^2\} |\nabla \eta|^2 \ dx. \end{align*} As for the second term, we can estimate it as \begin{align*} \int_\Omega a(1+|u|^2 \min\{|u|^{2s}, L^2\} \eta^2 \ dx = & \int_\Omega a \min\{|u|^{2s}, L^2\} \eta^2 \ dx \\ & + 2 \int_\Omega a |u|^2 \min\{|u|^{2s}, L^2\} \eta^2 \ dx \\ = & 3 \int_\Omega a|u|^2 \min\{|u|^{2s}, L^2\} \eta^2 \ dx \\ & + \int_\Omega a\min\{|u|^{2s}, L^2\} \eta^2 (1 - |u|^2) \ dx \\ \leq & 3 \int_\Omega a|u|^2 \min\{|u|^{2s}, L^2\} \eta^2 \ dx \\ & + \int_\Omega a \eta^2 \ dx. \end{align*} Therefore, \begin{align} \begin{split} & \int_\Omega |\nabla u|^2 \min\{|u|^{2s}, L^2\} \eta^2 \ dx + \frac s2 \int_{\{|u|^s\leq L \}} |\nabla(|u|^2)|^2 |u|^{2s - 2} \eta ^2 \ dx \\ & \qquad \leq \frac 12 \int_\Omega |\nabla u|^2 \min \{|u|^{2s}, L^2\} \eta^2 \ dx + c \int_\Omega u^2 \min\{|u|^{2s}, L^2\} |\nabla \eta|^2 \ dx \\ & \qquad \quad + 3 \int_\Omega a|u|^2 \min\{|u|^{2s}, L^2\} \eta^2 \ dx + \int_\Omega a \eta^2 \ dx. \end{split} \end{align}