arXiv 1811.033652018-11-08math.AP

具有不定号位势的奇异超线性薛定谔方程负能量解的多重性

Multiplicity of negative-energy solutions for singular-superlinear Schrödinger equations with indefinite-sign potential

Carlos Alberto Santos, Ricardo Alves Lima, Kaye Silva

摘要：

我们关心的是奇异超线性和亚临界薛定谔方程$$ \begin{array}{c} -Δu +V(x)u=λa(x)u^{-γ}+b(x)u^{p}~\mbox{in}~ \mathbb{R}^{N}, \end{array} $$的Nehari极值以外的正解的多重性，定义在Il'yasov [Topol.方法非线性分析，2017]，当电势$b \in L^{\infty}(\mathbb{R}^{N})$可以改变其符号时，$0<a\in L^{\frac{2}{1+γ}}(\mathbb{R}^{N})$，$V$是正连续函数，$N\geq 3$和$λ>0$是实参数。主要的困难来自于能量泛函的不可微性和Nehari集的连通分支的边界的交集非空的事实。我们克服了这些困难，探索边界的拓扑结构，建立非空的边界有空的交集，并通过控制能量水平最小化他们。

英文摘要：

We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schrödinger equation $$ \begin{array}{c} -Δu +V(x)u=λa(x)u^{-γ}+b(x)u^{p}~\mbox{in}~ \mathbb{R}^{N}, \end{array} $$ beyond the Nehari extremal value, as defined in Il'yasov [Topol. Methods Nonlinear Anal., 2017], when the potential $b \in L^{\infty}(\mathbb{R}^{N})$ may change its sign, $0<a\in L^{\frac{2}{1+γ}}(\mathbb{R}^{N})$, $V$ is a positive continuous function, $N\geq 3$ and $λ>0$ is a real parameter. The main difficulties come from the non-differentiability of the energy functional and the fact that the intersection of the boundaries of the connected components of the Nehari set is non empty. We overcome these difficulties by exploring topological structures of that boundary to build non-empty sets whose boundaries have empty intersection and minimizing over them by controlling the energy level.

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摘要：

英文摘要：

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