论文标题
非线性分数相对论schrödinger方程：存在，多样性，衰减和浓度结果
The nonlinear fractional relativistic Schrödinger equation: existence, multiplicity, decay and concentration results
论文作者
论文摘要
在本文中，我们研究了以下类别相对论schrödinger方程：\ begin {equation*} \ left \ {\ oken {array} {ll} {ll} {ll}（ - δ + m^{2}）^{s} \ Mathbb {r}^{n}，\\ u \ in H^{s}（\ Mathbb {r}^{n}），\ Quad U> 0＆\ mbox {in} \ Mathbb {inbb {r} \ end {equation*}其中$ \ varepsilon> 0 $是一个小参数，$ s \ in（0，1）$，$ m> 0 $，$ n> 2S $，$（ - δ+m^{2}） \ mathbb {r} $是满足当地条件的连续潜力，$ f：\ mathbb {r} \ rightarrow \ rightarrow \ mathbb {r} $是连续的亚临界非线性。通过使用扩展方法的变体和惩罚技术，我们首先证明，对于$ \ varepsilon> 0 $ hill，上述问题承认弱解决方案$ u _ {\ varepsilon} $，该$集中在本地最小点$ v $ as $ v $ as $ \ varepsilon \ varepsilon \ rightarrow 0 $ 0 $。我们还表明，$ u _ {\ varepsilon} $通过构建合适的比较功能并执行一些精致的估计，在Infinity中具有指数衰减。其次，通过将广义的nehari歧管方法和ljusternik-schnirelman理论相结合，我们将阳性解决方案的数量与套装的拓扑相关联，其中潜在的$ v $达到其最小值。
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.
