On some weakly coercive quasilinear problems with forcing

Abstract

We consider the forced problem $-{\Delta }_{p}u-V(x)|u{|}^{p-2}u=f(x)$, where ${\Delta }_p$ is the $p$-Laplacian ($1<p<\infty$) in a domain $\Omega \subset {\mathbb{R}}^N$, $V\ge 0$ and $Q_V(u):={\int }_{\Omega }|\nabla u{|}^{p}dx-{\int }_{\Omega }V|u{|}^{p}dx$ satisfies the condition (A) stated at the beginning of the paper. We show that this problem has a solution for all $f$ in a suitable space of distributions. Then we apply this result to some classes of functions $V$ which in particular include the Hardy potential and the potential $V(x)={\lambda }_{1,p}(\Omega )$, where ${\lambda }_{1,p}(\Omega )$ is the Poincaré constant on an infinite strip.