Bifurcation results and multiple solutions for the fractional $(p,q)$-Laplace operators

TL;DR

This paper studies a complex nonlocal eigenvalue problem involving fractional $(p,q)$-Laplace operators, establishing bifurcation phenomena and the existence of multiple solutions using variational techniques.

Contribution

It provides new bifurcation results and demonstrates the existence of multiple solutions for a fractional nonlocal eigenvalue problem involving $(p,q)$-Laplace operators.

Findings

Bifurcation from trivial solutions identified.

Bifurcation from infinity established.

Multiple solutions proven using variational methods.

Abstract

We investigate a nonlinear nonlocal eigenvalue problem involving the sum of fractional $(p,q)$-Laplace operators $(-\Delta)_p^{s_1}+(-\Delta)_q^{s_2}$ with $s_1,s_2\in (0,1)$; $p,q\in(1,\infty)$ and subject to Dirichlet boundary conditions in an open bounded set of $\mathbb{R}^N$. We prove bifurcation results from trivial solutions and from infinity for the considered nonlinear nonlocal eigenvalue problem. We also show the existence of multiple solutions of the nonlinear nonlocal problem using variational methods.