论文标题
具有渐近线性非线性的分数Schrodinger方程的多个半古典状态
Multiple semiclassical states for fractional Schrodinger equations with asymptotically linear nonlinearities
论文作者
论文摘要
在本文中，我们考虑了奇特的扰动分数schrödinger方程\ begin {equation*}ε^{2α}（ - δ） $α\ in（0,1）$，$（ - δ）^α$是订单$α$，$ v $具有全球最低点的分数Laplacian操作员，而在Infinity上是渐近线性的。我们调查了潜在的$ v $达到其全球最低最低限度的积极解决方案数量与集合拓扑之间的关系。如果$ v $有几个严格的全球最低点，我们还会构建多个集中解决方案。特别是，引入了一些新的技巧和Nehari歧管的方法，以克服因渐近线性非线性的出现而产生的困难。
In this paper, we consider the singularly perturbed fractional Schrödinger equation \begin{equation*} ε^{2α}(-Δ)^αu+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $ε>0$ is a small parameter, $α\in(0,1)$, $(-Δ)^α$ is the fractional Laplacian operator of order $α$, $V$ possesses global minimum points, and $f$ is asymptotically linear at infinity. We investigate the relationship between the number of positive solutions and the topology of the set where the potential $V$ attains its global minimum. We also construct multiple concentrating solutions if $V$ has several strict global minimum points. In particular, some new tricks and the method of Nehari manifold dependent on a suitable restricted set are introduced to overcome the difficulty resulting from the appearance of asymptotically linear nonlinearity.
