Existence of solutions for nonlinear equations with mixed local and nonlocal operators

TL;DR

This paper proves the existence of solutions for a nonlinear elliptic equation involving both local and nonlocal operators, using a critical point theorem, under certain conditions on parameters and source terms.

Contribution

It introduces a novel approach to establish solutions for equations with mixed local and nonlocal operators using Ricceri's critical point theorem.

Findings

Existence of solutions established for small parameters and specific source terms.

Application of Ricceri's theorem to mixed local-nonlocal operator equations.

Conditions under which solutions exist are characterized.

Abstract

We study an elliptic equation, with homogeneous Dirichlet boundary conditions, driven by a mixed type operator (the sum of the Laplacian and the fractional Laplacian), involving a parametric reaction and an undetermined source term. Applying a recent abstract critical point theorem of Ricceri, we prove existence of a solution for a convenient source and small enough parameters.