Multiplicity of positive solutions of a nonlinear Schrödinger equation

Yanheng Ding, Kazunaga Tanaka

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    Abstract

    We consider the multiple existence of positive solutions of the following nonlinear Schrödinger equation: -Δu + (λa(x) + b(x))u = u p, u > 0 in RN, where p ∈ (1, N+2/N-2) if N ≥ 3 and p ∈ (1, ∞) if N = 1, 2, and a(x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well Ω := int a-1 (0) consisting of k components Ω1, ... , Ωk and the first eigenvalues of -Δ + b(x) on Ω j under Dirichlet boundary condition are positive for all j = 1,2, ..., k. Under these conditions we show that (Pλ) has at least 2k - 1 positive solutions for large λ. More precisely we show that for any given non-empty subset J ⊂ {1, 2,...k},(Pλ.) has a positive solutions uλ(x) for large λ. In addition for any sequence λn → ∞ we can extract a subsequence λni; along which uλni converges strongly in H1 (RN). Moreover the limit function u(x) = limi → ∞ uλni satisfies (i) For j ∈ J the restriction u |Ωj of u(x) to Ωj is a least energy solution of -Δu + b(x)v = up in Ω j and u = 0 on ∂Ωj. (ii) u(x) = 0 for X ∈ RN \ (∪jεJ Ωj).

    Original languageEnglish
    Pages (from-to)109-135
    Number of pages27
    JournalManuscripta Mathematica
    Volume112
    Issue number1
    DOIs
    Publication statusPublished - 2003 Sep

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    ASJC Scopus subject areas

    • Mathematics(all)

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