Ground states of nonlinear Schr\"odinger equations with sum of periodic and inverse-square potentials
Ground states of nonlinear Schr\"odinger equations with sum of periodic and inverse-square potentials
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DOI：
10.1016/j.jde.2015.11.006
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发表时间：
2014-12
期刊：
影响因子：
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通讯作者：
Qianqiao Guo;Jarosław Mederski
中科院分区：
文献类型：
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作者：
Qianqiao Guo;Jarosław Mederski
研究了非线性薛定谔方程− Δ u+（V（x）− μ）解的存在性|X| 2）u = f（x，u），其中V：RN → R和f：RN × R → R在x ∈ RN中是周期的.我们假设0不在− Δ + V的谱中，且μ <（N − 2）2 4，N ≥ 3。超线性次临界项f满足弱单调性条件。对于足够小的μ ≥ 0，我们发现基态解是自然约束下能量泛函的极小化子。如果μ <0且0位于− Δ + V的谱线以下，则基态解不存在。
We study the existence of solutions of the following nonlinear Schrödinger equation− Δ u+(V (x)− μ| x| 2) u= f (x, u) for x∈ R N∖{0}, where V: R N→ R and f: R N× R→ R are periodic in x∈ R N. We assume that 0 does not lie in the spectrum of− Δ+ V and μ<(N− 2) 2 4, N≥ 3. The superlinear and subcritical term f satisfies a weak monotonicity condition. For sufficiently small μ≥ 0 we find a ground state solution as a minimizer of the energy functional on a natural constraint. If μ< 0 and 0 lies below the spectrum of− Δ+ V, then ground state solutions do not exist.