Three nontrivial solutions for nonlinear fractional Laplacian equations

TL;DR

This paper proves the existence of three nonzero solutions for a nonlinear fractional Laplacian boundary value problem using variational methods, spectral theory, and Morse theory, depending on the reaction term's growth.

Contribution

It introduces new results on multiple solutions for fractional Laplacian equations by applying advanced variational techniques tailored to sublinear and superlinear cases.

Findings

Established existence of three solutions for sublinear reactions using the second deformation theorem.

Proved the existence of solutions in the superlinear case via mountain pass and Morse theory.

Demonstrated the effectiveness of spectral theory in analyzing fractional Laplacian problems.

Abstract

We study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three nonzero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.