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

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 $W_0^{s_1,p}(\Omega )\hookrightarrow L^{\infty }\left(\Omega \right)$ failing, we establish $\Vert u{\Vert }_{L^{\infty }\left(\Omega \right)}\le C[u{]}_{s_1,p}$ (for some $C>0$ ), when $u$ is a solution.