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

In this paper, we are interested in studying the multiplicity, uniqueness, and nonexistence of solutions for a class of singular elliptic eigenvalue problems for the Dirichlet fractional (p, q)-Laplacian. The nonlinearity considered involves supercritical Sobolev growth. Our approach is variational together with the sub- and supersolution methods, and in this way we can address a wide range of problems not yet contained in the literature. Even when \(W^{s_1,p}_0(\Omega ) \hookrightarrow L^{\infty }\left( \Omega \right) \) $W_0^{s_1,p}(\mathrm{\Omega })\hookrightarrow L^{\infty }\left(\mathrm{\Omega }\right)$ failing, we establish \(\Vert u\Vert _{L^{\infty }\left( \Omega \right) } \le C[u]_{s_1,p}\) ${\Vert u\Vert }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le C{[u]}_{s_1,p}$ (for some \(C>0\) $C>0$ ), when u is a solution.