On an Anisotropic $p$-Laplace equation with variable singular exponent

Abstract

 −∆H,pu = λf(x) uq(x) + g(u) in Ω, u > 0 in Ω, u = 0 on ∂Ω, under the assumption Ω is a bounded smooth domain in R with p,N ≥ 2, λ > 0 and 0 < q ∈ C(Ω̄). For the purely singular case that is g ≡ 0, we proved existence and uniqueness of solution. We also demonstrate the existence of multiple solution to (P ) provided f ≡ 1 and g(u) = u for r ∈ (p− 1, p − 1).