The Discussion for the Existence of Nontrivial Solutions About a Kind of Quasi-Linear Elliptic Equations

摘要

The paper is concerned with the follow problems: { -div(｜x｜~α｜▽u｜~(p-2)▽u) = ｜x｜~βu~(p(α,β)-1) - λ｜x｜~γu~(p-1) + ｜x｜~μu~(q-1) u(x) ＞ 0,x ∈Ω ｜▽u｜~(p-2)(б)u/(б)n=0 x∈(б)Ω It is the kind of the problem with Neumann boundary. Let Ω be a bounded domain with a smooth C~2 boundary in R~N(N＞3), 0 ∈Ω, and n denote the unit outward normal to (б)Ω, and 1 ＜p＜N, and α＜0, β＜0, such that P(α,β)△=p(N+β)/N-p-α ＞P,γ＞α-P, p＜q＜p(α,μ), For various parameters α,β, γ and μ, we establish some existence results of the solutions in the case of 0 ∈Ω. The novelty of the paper is that the scopes of the parameters α, β, γ and μ, play an important role in the existence of solutions, because of the singularity and the non-compactness.