L∞ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity

Abstract

We present new L∞ 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 −Δu=f(x,u), in Ω, with Dirichlet boundary conditions, where Ω⊂RN, with N>2, is a bounded smooth domain, and f is a subcritical Carathéodory non-linearity. We provide L∞ a priori estimates for weak solutions, in terms of their L2∗-norm, where 2∗=2N/N−2 is the critical Sobolev exponent. By a subcritical non-linearity we mean, for instance, |f(x,s)|≤|x|−μ˜f(s), where μ∈(0,2), and ˜f(s)/|s|2∗μ−1→0 as |s|→∞, here 2∗μ:=2(N−μ)/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−μ |s|2∗μ−2s/[log(e+|s|)]β, with μ∈[1,2), then, for any ε>0 there exists a constant Cε>0 such that for any solution u∈H10(Ω), the following holds [log(e+∥u∥∞)]β≤Cε(1+∥u∥2∗)(2∗μ−2)(1+ε).