Existence Results for Quasilinear Elliptic Equations with Indefinite Weight

Abstract EN

We provide the existence of a solution for quasilinear elliptic equation -div(a∞(x)|∇u|p-2∇u+ã(x,|∇u|)∇u)=λm(x)|u|p-2u+f(x,u)+h(x) in Ω under the Neumann boundary condition.

Here, we consider the condition that ã(x,t)=o(tp-2) as t→+∞ and f(x,u)=o(|u|p-1) as |u|→∞.

As a special case, our result implies that the following p-Laplace equation has at least one solution: -Δpu=λm(x)|u|p-2u+μ|u|r-2u+h(x) in Ω,∂u/∂ν=0 on ∂Ω for every 1

Moreover, in the nonresonant case, that is, λ is not an eigenvalue of the p-Laplacian with weight m, we present the existence of a solution of the above p-Laplace equation for every 1