Existence results for a class of local and nonlocal nonlinear elliptic problems

Abstract

In this paper, we study the $p$-Laplacian problems in the case where $p$ depends on the solution itself. We consider two situations, when $p$ is a local and nonlocal quantity. By using a singular perturbation technique, we prove the existence of weak solutions for the problem associated to the following equation $$ \begin{cases} -\mathrm{d}\mathrm{i}\mathrm{v}(|\nabla u|^{p(u)-2}\nabla u)+|u|^{p(u)-2}u=f & \mbox{in}\; \Omega\\ u=0& \mbox{on}\; \partial\Omega , \end{cases}$$ and also for its nonlocal version. The main goal of this paper is to extend the results established by M. Chipot and H. B. de Oliveira (Math. Ann., 2019, 375, 283-306).