Multiplicity results for non-local operators of elliptic type

TL;DR

This paper establishes multiple solutions for a class of non-local elliptic problems using variational methods and Morse theory, revealing solution multiplicity depending on the spectral parameter and nonlinear symmetry.

Contribution

It introduces new multiplicity results for non-local operators of elliptic type, employing critical group analysis and Morse theory to extend existing solution existence theories.

Findings

Three solutions for λ < λ₁

Two solutions for λ ≥ λ₁

Infinite solutions when f is odd and λ ≥ λ₁

Abstract

In this paper, we study a class of problems proposed by Servadei and Valdinoci in \cite{Ser3}; namely, \begin{equation}\label{prob_0} \left\{\begin{aligned} -\mathcal{L}_{K} u(x)-\lambda u(x) & =f(x,u), \mbox{ for } x\in \Omega; u & =0 \quad \text{ in } \mathbb{R}^{N}\backslash\Omega, \end{aligned} \right. \end{equation} where $\Omega\subset \mathbb{R}^{N}$ is an open bounded set with Lipschitz boundary, $\lambda\in\mathbb{R}$, $f\in C^{1}(\overline{\Omega}\times\mathbb{R},\mathbb{R})$, with $f(x,0) = 0$ for $x\in\Omega$, and $\mathcal{L}_K$ is a non-local integrodifferential operator with homogeneous Dirichlet boundary condition. By computing the critical groups of the associated energy functional for problem (1) at the origin and at infinity, respectively, we prove that problem (\ref{prob_0}) has three nontrivial solutions for the case $\lambda < \lambda_1$ and two nontrivial solutions for the case $\lambda\geqslant\lambda_1,$ where $\lambda_1$ is the first eigenvalue of the operator $-\mathcal{L}_K$. Finally, assuming that the nonlinearity $f$ is odd in the second variable, we prove the existence of an unbounded sequence of weak solutions of problem (1) for the case $\lambda\geqslant\lambda_1$. We use variational methods and infinite-dimensional Morse theory to obtain the results.