Free boundary solutions to a log-singular elliptic equation

Abstract

The aim of this paper is study the equation −Δu=(log (u)+λu^p)χ_{u>0} in Ω with Dirichlet boundary condition, where 0<p<(N+2)/(N−2) and p≠1. We regularize the term log (u) for u near 0 by using a function g_ε(u)=−log ((u²+εu+ε)/(u+ε)) for u≥0 which tends to log (u) as ε→0 pointwisely. When the parameter λ>0 is sufficiently large, the corresponding energy functional to the perturbed equation −Δu+g_ε(u)=λ(u⁺)^p has nontrivial critical points u_ε in H₀¹(Ω). Letting ε→0, then u_ε converges to a solution of the original problem, which is nontrivial and nonnegative. For 1<p<(N+2)/(N−2) there is at least one nontrivial solution. While for 0<p<1, there are at least two nontrivial distinct solutions.