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具周期势与消失势之和的变号非线性薛定谔方程

Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities

Bartosz Bieganowski, Jarosław Mederski

发表信息：
Commun. Pure Appl. Anal., Vol. 17, Issue 1, (2018), p. 143-161
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摘要：

我们寻找下面的非线性薛定谔方程$$-Δu + V(x)u = f(x,u)-Γ(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$的基态解，其中$V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$是周期势$V_{per}$和定域势$V_{loc}$的和，$Γ\in L^{\infty}(\mathbb{R}^N)$是周期的，$Γ(x)\geq 0$对于a.e. $x\in\mathbb{R}^N$和$2\leq q<2^*$。我们假设$\infσ(-Δ+V)>0$，其中$σ(-Δ+V)$代表$-Δ+V$的谱，$f$具有亚临界增长但高于$Γ(x)|u|^{q-2}u$，然而非线性$f(x,u)-Γ(x)|u|^{q-2}u$可能改变符号。虽然一个Nehari型单调性条件的非线性不满足，我们调查的存在性的基态解是极小的Nehari流形。

英文摘要：

We look for ground state solutions to the following nonlinear Schrödinger equation $$-Δu + V(x)u = f(x,u)-Γ(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is the sum of a periodic potential $V_{per}$ and a localized potential $V_{loc}$, $Γ\in L^{\infty}(\mathbb{R}^N)$ is periodic and $Γ(x)\geq 0$ for a.e. $x\in\mathbb{R}^N$ and $2\leq q<2^*$. We assume that $\infσ(-Δ+V)>0$, where $σ(-Δ+V)$ stands for the spectrum of $-Δ+V$ and $f$ has the subcritical growth but higher than $Γ(x)|u|^{q-2}u$, however the nonlinearity $f(x,u)-Γ(x)|u|^{q-2}u$ may change sign. Although a Nehari-type monotonicity condition for the nonlinearity is not satisfied we investigate the existence of ground state solutions being minimizers on the Nehari manifold.

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