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

Antonio Iannizzotto

TL;DR

The paper addresses the existence of solutions to a nonlinear elliptic equation with a mixed local (Laplacian) and nonlocal (fractional Laplacian) operator under Dirichlet conditions. It develops a variational framework on appropriate Sobolev spaces and localizes solutions within the unit ball of the fractional seminorm by leveraging Ricceri's abstract critical point theorem. By constructing suitable energy functionals and verifying the theorem's hypotheses, it proves the existence of a weak solution for small λ and a smooth source h, with localization [u]_s < 1. A concrete corollary for pure power nonlinearities demonstrates the applicability of the method to subcritical regimes.

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.

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

TL;DR

The paper addresses the existence of solutions to a nonlinear elliptic equation with a mixed local (Laplacian) and nonlocal (fractional Laplacian) operator under Dirichlet conditions. It develops a variational framework on appropriate Sobolev spaces and localizes solutions within the unit ball of the fractional seminorm by leveraging Ricceri's abstract critical point theorem. By constructing suitable energy functionals and verifying the theorem's hypotheses, it proves the existence of a weak solution for small λ and a smooth source h, with localization [u]_s < 1. A concrete corollary for pure power nonlinearities demonstrates the applicability of the method to subcritical regimes.

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.

Paper Structure

This paper contains 3 sections, 4 theorems, 41 equations.

Table of Contents

  1. Introduction and main result
  2. Background
  3. Proof of the main result

Key Result

Theorem 1.1

Let $\Omega\subset{\mathbb R}^N$ ($N\geqslant 2$) be a bounded domain with a $C^1$ boundary, $r\in(1,2^*)$. Then, there exists $h\in C^\infty_c(\Omega)$ with the following property: for all $q\in(1,2^*)$ there exists $\lambda^*>0$ s.t. for all $\lambda\in[0,\lambda^*]$ problem log has at least one s

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • proof