Eigenvalue for a problem involving the fractional (p,q)-Laplacian operator and nonlinearity with a singular and a supercritical Sobolev growth

TL;DR

This paper investigates the eigenvalue problem involving the fractional (p,q)-Laplacian with nonlinearities exhibiting supercritical Sobolev growth, establishing solution properties using variational and sub-supersolution methods.

Contribution

It introduces a novel approach combining variational techniques with sub- and supersolution methods for a broad class of fractional (p,q)-Laplacian problems with supercritical growth.

Findings

Established multiplicity, uniqueness, and nonexistence results.

Derived bounds for solutions even when Sobolev embeddings fail.

Extended the literature to include problems with supercritical nonlinearities.

Abstract

In this paper, we are interested in studying the multiplicity, uniqueness, and nonexistence of solutions for a class of singular elliptic eigenvalue problem for the Dirichlet fractional $(p,q)$-Laplacian. The nonlinearity considered involves supercritical Sobolev growth. Our approach is variational togheter with the sub- and supesolution methods, and in this way we can address a wide range of problems not yet contained in the literature. Even when $W^{s_1,p}_0(\Omega) \hookrightarrow L^{\infty}\left(\Omega\right)$ failing, we establish $\|u\|_{L^{\infty}\left(\Omega\right)} \leq C[u]_{s_1,p}$ (for some $C>0$ ), when $u$ is a solution.