Multiplicity of positive solutions of a nonlinear Schrödinger equation
Multiplicity of positive solutions of a nonlinear Schrödinger equation
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DOI：
10.1007/s00229-003-0397-x
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发表时间：
2003-08
影响因子：
0.6
通讯作者：
Yanheng Ding;Kazunaga Tanaka
中科院分区：
文献类型：
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作者：
Yanheng Ding;Kazunaga Tanaka
我们考虑如下非线性薛定谔方程正解的多重存在性：其中p∈(1，N+2N-2)当N≥3时，p(1，∞)当N=1，2时，a(X)，b(X)是连续函数。我们假设a(X)是非负的，并且有一个势井Ω：=int由k个分量构成的−1(0)，在Dirichlet边界条件下，−Δ+b(X)在Ωj上的第一本征值对所有j=1，2，...，k都是正的.在这些条件下，我们证明了(PMλ)对于大−至少有2k个λ1正解.更确切地说，我们证明了对于任何给定的非空子集J⊂{1，2，…k\}，(Pλ)对于大的λ(X)有一个正解uλ(X)。此外，对于任意序列λn→∞，我们都可以得到一个子序列λn i，沿着它uλn i在H1(RN)中强收敛.此外，极限函数u(X)=Lim i→∞uλni满足(I)对于jJ，u(X)对Ωj的限制u|Ωj是−Δv+b(X)v=Vp在Ωj上且v=0在∂Ωj上的最小能量解.(Ii)u(X)=0对x∈\bfR^N∖(j∈J\omega_j).
We consider the multiple existence of positive solutions of the following nonlinear Schrödinger equation: where p∈(1,N+2N-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 \Omega_1,...,\Omega_k 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 (PM λ) has at least 2 k− 1 positive solutions for large λ. More precisely we show that for any given non-empty subset J⊂{1,2,...k\},(P λ) has a positive solutions u λ (x) for large λ. In addition for any sequence λ n→∞ we can extract a subsequence λ ni along which u λni converges strongly in H 1 (RN). Moreover the limit function u (x)= lim i→∞ u λni satisfies (i) For j J the restriction u| Ω j of u (x) to Ω j is a least energy solution of− Δ v+ b (x) v= vp in Ω j and v= 0 on∂ Ω j.(ii) u (x)= 0 for x∈\bfR^N∖(j∈J\Omega_j).