Nonlinear Dirichlet Problem with Non Local Regional Diffusion

Abstract

Abstract The purpose of this paper is to study the existence of solutions for equations driven by a non-local regional operator with homogeneous Dirichlet boundary conditions. More precisely, we consider the problem ( − Δ ) ρ α u ( x ) = f ( x , u ) , x ∈ Ω , u ( x ) = 0 , x ∈ ∂ Ω $$\begin{array}{c}{(-\Delta )_{\rho }^{\alpha }u(x)=f(x, u),\quad x\in \Omega,}\cr {u(x)=0, \quad x\in \partial \Omega }\end{array}$$ where the nonlinear term f satisfies superlinear and subcritical growth conditions at zero and at infinity. These equations have a variational structure, and so its solutions can be found as critical points of the energy functional Iρ associated to the problem. Here we get such critical pointsusing the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary.