Semilinear elliptic equations with Hardy potential and subcritical source term

Abstract

Let \(\Omega \) be a smooth bounded domain in \({\mathbb {R}}^N\) (\(N>2\)) and \(\delta (x):=\text {dist}\,(x,\partial \Omega )\). Assume \(\mu \in {\mathbb {R}}_+, \nu \) is a nonnegative finite measure on \(\partial \Omega \) and \(g \in C(\Omega \times {\mathbb {R}}_+)\). We study positive solutions of $$\begin{aligned} -\Delta u - \frac{\mu }{\delta ^2} u = g(x,u) \text { in } \Omega , \qquad \text {tr}^*(u)=\nu . \end{aligned}$$ (P) Here \(\text {tr}^*(u)\) denotes the normalized boundary trace of u which was recently introduced by Marcus and Nguyen (Ann Inst H Poincare Anal Non Lineaire, 34, 69–88, 2017). We focus on the case \(0 0\), we prove that there is a critical value \(q^*\) (depending only on \(N, \mu \)) for (P) in the sense that if \(q<q^*\) then (P) possesses a solution under a smallness assumption on \(\nu \), but if \(q \ge q^*\) this problem admits no solution with isolated boundary singularity. Existence result is then extended to a more general setting where g is subcritical [see (1.28)]. We also investigate the case where g is linear or sublinear and give an existence result for (P).