Existence Results for Quasilinear Elliptic Equations with Indefinite Weight

Abstract

We provide the existence of a solution for quasilinear elliptic equation −div(𝑎∞(𝑥)|∇𝑢|𝑝−2∇𝑢+̃𝑎(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 ̃𝑎(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<𝑟<𝑝<∞, 𝜆∈ℝ, 𝜇≠0 and 𝑚,ℎ∈𝐿∞(Ω) with ∫Ω𝑚𝑑𝑥≠0. 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<𝑟<𝑝<∞, 𝜇∈ℝ and 𝑚,ℎ∈𝐿∞(Ω).