Multiple Weak Solutions for a Class of Singular Equations Involving the p(y)-Laplace Operator

Abstract

In this work, we are interested in the multiple weak solutions for the following singular elliptic problem involving the p(y)-Laplacian with the Dirichlet boundary condition: -∆_p(y) w + m(y) |w|^(p(y)-2) w=g(y) |w|^(ξ(y)-2) w+(λ h(y))/w^η(y) in Ω, w > 0 in Ω, w = 0 on ∂Ω.

Here, the operator ∆_p(y) w = div(|∇w|^(p(y)-2) ∇w) represents the p(y)-Laplace operator, where p(y) is a non-constant continuous function. The domain Ω ⊂ R^N (N ≥2) is bounded with a C^2-boundary, and λ is a positive parameter. The function m(y) is positive, while g(y), h(y) ∈ C(Ω) are non-negative weight functions with compact support in Ω. Additionally, η(y), p(y), ξ(y) ∈ C(Ω) satisfy some appropriate conditions. We apply the Nehari manifold and fibering map method to establish the existence and multiplicity of positive weak solutions.