$L^{\infty }$ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity

Abstract

We present new $L^{\infty }$ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in combining elliptic regularity with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a semilinear boundary value problem $-\Delta u=f(x,u),$ in $\Omega ,$ with Dirichlet boundary conditions, where $\Omega \subset {\mathbb{R}}^N$, with $N>2,$ is a bounded smooth domain, and $f$ is a subcritical Carathéodory non-linearity. We provide $L^{\infty }$ a priori estimates for weak solutions, in terms of their $L^{2^{\ast }}$-norm, where $2^{\ast }=\frac{2N}{N-2}$ is the critical Sobolev exponent. By a subcritical non-linearity we mean, for instance, $|f(x,s)|\le |x{|}^{-\mu }\tilde{f}(s),$ where $\mu \in (0,2),$ and $\tilde{f}(s)/|s{|}^{2_{\mu }^{\ast }-1}\to 0$ as $|s|\to \infty$, here $2_{\mu }^{\ast }:=\frac{2(N-\mu )}{N-2}$ is the critical Sobolev-Hardy exponent. Our non-linearities includes non-power non-linearities. In particular we prove that when $f(x,s)=|x{|}^{-\mu }\frac{|s{|}^{2_{\mu }^{\ast }-2}s}{[\log (e+|s|){]}^{\beta }},$ with $\mu \in [1,2),$ then, for any $\epsilon >0$ there exists a constant $C_{\epsilon }>0$ such that for any solution $u\in H_0^1(\Omega )$, the following holds $$\left[\log \right(e+\Vert u{\Vert }_{\infty }){]}^{\beta }\le C_{\epsilon }(1+\Vert u{\Vert }_{2^{\ast }}{)}^{(2_{\mu }^{\ast }-2)(1+\epsilon )}.$$