$L^\infty$ a-priori estimates for subcritical semilinear elliptic equations with a Carath\'eodory 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 Carath\'eodory non-linearity. We provide $L^\infty$ a priori estimates for weak solutions, in terms of their $L^{2^*}$-norm, where $2^*=\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 $\tilde{f}(s)/|s|^{2_{\mu}^*-1}\to 0$ as $|s|\to \infty$, here $2^*_{\mu}:=\frac{2(N-\mu)}{N-2}$ is the critical Sobolev-Hardy exponent. Our non-linearities includes non-power non-linearities. In particular we prove that when $f(x,s)=|x|^{-\mu}\,\frac{|s|^{2^*_{\mu}-2}s}{\big[\log(e+|s|)\big]^\beta}\,,$ with $\mu\in[1,2),$ then, for any $\varepsilon>0$ there exists a constant $C_\varepsilon>0$ such that for any solution $u\in H^1_0(\Omega)$, the following holds $$ \Big[\log\big(e+\|u\|_{\infty}\big)\Big]^\beta\le C _\varepsilon \, \Big(1+\|u\|_{2^*}\Big)^{\, (2^*_{\mu}-2)(1+\varepsilon)}\, . $$