# Brezis-Kato regularity argument - Some questions about Struwe's proof

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 begins 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 $$\int_\Omega |\nabla u|^2 \min\{|u|^{2s}, L^2\} \eta^2 \ dx + \frac s2 \int_{\{|u|^s\leq L \}} |\nabla(|u|^2)|^2 |u|^{2s - 2} \eta ^2 \ dx \leq \\ -2 \int_\Omega \nabla u u \min \{|u|^{2s}, L^2\} \nabla \eta \eta \ dx + \int_\Omega a(1 + 2|u|^2)\min \{|u|^{2s}, L^2\}\eta^2 \ dx.$$

Why is $$\varphi \in H_0^1(\Omega)$$? How do the second term in the left-hand side of the inequality arises?

I tried the following: We want to compute

$$\int_{\{|u|^s \leq L\}} \nabla u u \nabla |u|^{2s} \eta^2 \ dx .$$ But $$\nabla |u|^{2s} = \nabla(u^+ - u^-)^{2s} = 2s |u|^{2s - 1} \nabla |u|$$ so we get $$\int_{\{|u|^s \leq L\}} \nabla u u \nabla |u|^{2s} \eta^2 \ dx = 2s \int_{\{|u|^s \leq L\}} (\nabla u \nabla |u|) u |u|^{2s - 1} \eta ^2 \ dx$$ On the other hand, $$\frac s2 \int_{\{|u|^s \leq L\}} |\nabla |u|^2|^2 |u|^{2s - 2} \eta^2 \ dx = \frac s2 \int_{\{|u|^s \leq L\}} |2 |u| \nabla |u||^2 |u|^{2s - 2} \eta^2 \ dx \\ = 2s \int_{\{|u|^s \leq L\}} |\nabla |u||^2 |u|^{2s} \eta ^2 \ dx.$$ How to conclude that these two expressions are the same?

Also, what is the intuition for the proof of this theorem? It is seeming like just a lot of calculations.

Thanks in advance and kind regards.

The function $$f\colon \mathbb{R}\to\mathbb{R},\,\lambda\mapsto \lambda \min\{\lvert \lambda\rvert^{2s},L^2\}$$ is Lipschitz, thus $$f\circ u\in H^1_{loc}(\Omega)$$. It's not super fun to prove that; it's clear for smooth $$f$$ with bounded derivative, and then you have to do some approximation. Anyway, this implies $$\phi=\eta^2 (f\circ u)\in H^1_0(\Omega)$$ -- just use the product rule and use that $$\phi$$ has compact support.
As for the second part, your almost there: $$\nabla u=\mathrm{sgn}(u)\nabla \lvert u\rvert$$, you just have to pull $$\mathrm{sgn}(u)$$ from the gradient to the factor $$u$$ to get $$\lvert u\rvert$$ there.