Let $\Omega$ be a bounded open domain in $\mathbb{R}^n$ and let $u\in H_0^1(\Omega)$ be a weak solution of $$-\Delta u = |u|^{p-1}u\quad \text{in } \Omega$$ $$u = 0\quad\text{on }\partial\Omega$$ where $1<p<\frac{n+2}{n-2}$. How to prove that $u\in L^{\infty}(\Omega)$? I need some hits.
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2I think you need some regularity of the domain? Assuming this, you can use the $L^q$ ($1<q<\infty$) regularity for Poisson's equation: $\Delta u=f\in L^q$ implies $u\in W^{2,q}$. Your assumptions give $f=|u|^{p-1}u\in L^q$ for $q>2n/(n+2)$, which is exactly enough for this $W^{2,q}$ estimate, via Sobolev embedding, to land you $u\in L^r$ for some $r>2n/(n-2)$ (better than Sobolev for $H^1$). Then iterate this until you're bounded. – Jose27 Oct 19 '22 at 07:39