Positive Solutions To Quasilinear p(y )−Laplacian Equation With Robin Boundary Condition

Abstract

This work addresses the following Robin boundary value problem: -div(b(y)|∇w|^(p(y)-2)∇w) = λF(y) |w|^(η(y)-2) w+G(y) |w|^(γ(y)-2) w in Ω, b(y)|∇w|^(p(y)-2) ∂w/∂ϑ+ H(y) |w|^(p(y)-2) w=0 on ∂Ω,

where Ω ⊂ R^N (N ≥2) is a bounded domain, λ>0,H ∈C(∂Ω) and F, G ∈C(Ω ̅) are non-negative weight functions which has compact support in Ω. The function b(y) ∈ C^(0,δ)(Ω ̅) ∩ L^∞ (Ω ̅) is a positive. In addition, the functions p(y), η(y), γ(y) ∈C(Ω ̅)satisfy some appropriate conditions. We establish the existence and multiplicity of weak solutions to the considered problem in generalized Sobolev spaces W^(1,p(y) ) (Ω) by employing the Nehari manifold method.