Three Solutions for a Perturbed Integral Equation with Homogeneous Dirichlet Condition

Abstract

We consider the integral equation $-\mathcal{L}_K^p u = \lambda f(x,u) + \mu g(x,u),$ with homogeneous Dirichlet condition on a bounded Lipschitz domain of $\mathbb{R}^N$ where $\lambda, \mu \in \mathbb{R},$ $p \geq 2,$ $s \in ]0,1[,$ $N > ps,$ $f,g : \mathbb{R}^N \rightarrow \mathbb{R}$ are Carathéodory functions with subcritical growth and $-\mathcal{L}^p$ denotes a class of operators that includes $(-\Delta)_p^s,$ the fractional $p$-Laplacian. Here $\mu g$ represents a small perturbation of $\lambda f.$ Applying an abstract critical point theorem due to Ricceri, a variational setting developed by Xiang et al. and a Minti-Browder's theorem, we prove the existence of three weak solutions.