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arXiv 2003.06155math.AP
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非线性分数阶相对论性薛定谔方程：存在性、多重性、衰变和浓度结果

The nonlinear fractional relativistic Schrödinger equation: existence, multiplicity, decay and concentration results

Vincenzo Ambrosio

发表信息：
Discrete Cont. Dyn. Syst. (2021)
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摘要：

本文研究了如下一类分数阶相对论薛定谔方程：\begin{equation*} \left\{ \begin{array}{ll} (-Δ+m^{2})^{s}u + V(\varepsilon x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*}其中$\varepsilon>0$是小参数，$s\in (0, 1)$，$m>0$，$N> 2s$，$(-Δ+m^{2})^{s}$是分数阶相对论薛定谔算子，$V:\mathbb{R}^{N}\rightarrow \mathbb{R}$是满足局部条件的连续势，$f:\mathbb{R}\rightarrow \mathbb{R}$是连续次临界非线性项.通过使用一个变体的扩展方法和惩罚技术，我们首先证明，对于$\varepsilon>0$足够小，上述问题承认一个弱解$u_{\varepsilon}$集中在一个局部极小点的$V$作为$\varepsilon\rightarrow 0$。我们还表明，$u_{\varepsilon}$有一个指数衰减在无穷远，通过构建一个合适的比较功能，并通过执行一些改进的估计。其次，结合推广的Nehari流形方法和Ljusternik-Schnirelman理论，我们将正解的个数与位势PDMATHAMV达到最小值的集合的拓扑联系起来.

英文摘要：

In this paper we study the following class of fractional relativistic Schrödinger equations: \begin{equation*} \left\{ \begin{array}{ll} (-Δ+m^{2})^{s}u + V(\varepsilon x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $m>0$, $N> 2s$, $(-Δ+m^{2})^{s}$ is the fractional relativistic Schrödinger operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential satisfying a local condition, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous subcritical nonlinearity. By using a variant of the extension method and a penalization technique, we first prove that, for $\varepsilon>0$ small enough, the above problem admits a weak solution $u_{\varepsilon}$ which concentrates around a local minimum point of $V$ as $\varepsilon\rightarrow 0$. We also show that $u_{\varepsilon}$ has an exponential decay at infinity by constructing a suitable comparison function and by performing some refined estimates. Secondly, by combining the generalized Nehari manifold method and Ljusternik-Schnirelman theory, we relate the number of positive solutions with the topology of the set where the potential $V$ attains its minimum value.

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