Singular perturbations for superlinear problems in non-reflexive fractional Orlicz–Sobolev spaces

Abstract

Abstract In this work, we investigate the existence and multiplicity as well as regularity of solutions for a huge class of concave–convex problems involving the fractional upper Phi $\Phi$ Φ -Laplacian operator in non-reflexive setting. The main difficulty arises from the lack of homogeneity for the upper Phi $\Phi$ Φ -Laplacian operator. An additional difficulty stems from the interplay between a singular term and a superlinear term. To address these issues, we introduce a suitable class of upper N $N$ N -functions and employ the Nehari manifold method together with a nonlinear Rayleigh quotient.