Two positive solutions of a class of Schrödinger–Poisson system with indefinite nonlinearity
Two positive solutions of a class of Schrödinger–Poisson system with indefinite nonlinearity
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DOI：
10.1016/j.jde.2013.06.022
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发表时间：
2013-10
影响因子：
2.4
通讯作者：
Lirong Huang;E. Rocha;Jianqing Chen
中科院分区：
文献类型：
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作者：
Lirong Huang;E. Rocha;Jianqing Chen
我们研究了一类Schrödinger-Poisson系统的正解的存在性和多重性：{−Δ u+ u+ l (x) φ u= k (x)| u| p−2 u+ μ h (x) u在R 3中，−Δ φ = l (x) u 2在R 3中，其中4< p< 6, k∈C (R 3)， k在R 3中变号，lim| x|→∞k (x)= k∞< 0。我们主要证明了在μ> μ 1和接近μ 1的情况下存在至少两个正解，其中μ 1是- Δ+ i d在h1 （R 3）中的第一个特征值，其权函数H，其对应的特征函数用e1表示。一个有趣的现象是，我们不需要∫r3k (x) e1pdx < 0这个条件，而这个条件已被证明是具有不定非线性的半线性椭圆方程的正解存在的必要条件（参见S. Alama， G. Tarantello， On On On半线性椭圆方程具有不定非线性，Calc. Var.偏微分方程1(1993)439-475）。
We study the existence and multiplicity of positive solutions of a class of Schrödinger–Poisson system:{− Δ u+ u+ l (x) ϕ u= k (x)| u| p− 2 u+ μ h (x) u in R 3,− Δ ϕ= l (x) u 2 in R 3, where 4< p< 6, k∈ C (R 3), k changes sign in R 3 and lim| x|→∞ k (x)= k∞< 0. We mainly prove the existence of at least two positive solutions in the case that μ> μ 1 and near μ 1, where μ 1 is the first eigenvalue of− Δ+ i d in H 1 (R 3) with weight function h, whose corresponding eigenfunction is denoted by e 1. An interesting phenomenon is that we do not need the condition∫ R 3 k (x) e 1 p d x< 0, which has been shown to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see eg S. Alama, G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993) 439–475).