$\mathbb R^{N}$中分数阶薛定谔方程的Ljusternick-Schnirelmann范畴和莫尔斯理论的多重性结果
A multiplicity result via Ljusternick-Schnirelmann category and morse theory for a fractional schrödinger equation in $\mathbb R^{N}$
摘要：
本文研究了$\mathbb R^{N}, N>2s$ $$ \varepsilon^{2s} (-Δ)^{s}u + V(z)u=f(u), \,\,\, u(z) > 0 $$中的下列问题：其中$0<s<1$，$(-Δ)^{s}$是分数拉普拉斯算子，$\varepsilon$是正参数，势函数$V:\mathbb{R}^N \to\mathbb{R}$ %是连续函数，非线性函数$f:\mathbb R \to \mathbb R$满足适当的假设;特别地，假设$V$在某个集合$M.$上达到其正极小值。通过使用变分方法，我们证明了存在性，当$\varepsilon\to 0^{+}$ .特别是多重性的结果得到的Ljusternick-Schnirelmann和莫尔斯理论的装置，通过利用“拓扑复杂性”的集合$M$。
英文摘要：
In this work we study the following class of problems in $\mathbb R^{N}, N>2s$ $$ \varepsilon^{2s} (-Δ)^{s}u + V(z)u=f(u), \,\,\, u(z) > 0 $$ where $0<s<1$, $(-Δ)^{s}$ is the fractional Laplacian, $\varepsilon$ is a positive parameter, the potential $V:\mathbb{R}^N \to\mathbb{R}$ %is a continuous functions and the nonlinearity $f:\mathbb R \to \mathbb R$ satisfy suitable assumptions; in particular it is assumed that $V$ achieves its positive minimum on some set $M.$ By using variational methods we prove existence, multiplicity and concentration of maxima of positive solutions when $\varepsilon\to 0^{+}$. In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the "topological complexity" of the set $M$.
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