MULTIPLE POSITIVE SOLUTIONS FORTHE N-LAPLACE EQUATION WITHNONLINEAR NEUMANN BOUNDARY CONDITIONS

摘要

Let Ω ? R~Nbe a bounded domain with C~2boundary. Letu ∈ W~(1,N)(Ω) be a weak solution of the following problem: { -div(|?u|~(N-2)?_u) +|u|~(N-2)u= μh(u)e~u~α|?u|~(N-2)?u/?v=λψu~qon?Ω where α ∈ (0, N/N-1], λ, μ > 0, q ∈ [0,N — 1) and ψis a positive Holder continuous function on Ω. Here, h(u) is a "suitable" perturbation of e~u~αas u → ∞ (see assumptions (A1)-(A5) in Section 1). In this article,we show that there exists a region R ? {(μ, λ) : μ, λ > 0} bounded bythe graph of a map A such that (P_(μ,λ))admits at least two solutions forall (μ, λ) ∈R, at least one solution for any (μ, λ)∈ ?Rand no solutionfor (μ, λ) outside R.