Existence of stable standing waves for the nonlinear Schrödinger equation with attractive inverse-power potentials

Abstract

<abstract><p>In this paper, we consider the following nonlinear Schrödinger equation with attractive inverse-power potentials</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\partial_t\psi+\Delta\psi+\gamma|x|^{-\sigma}\psi+|\psi|^\alpha\psi = 0, \; \; \; (t, x)\in\mathbb{R}\times\mathbb{R}^N, $\end{document} </tex-math></disp-formula></p> <p>where $ N\geq3 $, $ 0 &lt; \gamma &lt; \infty $, $ 0 &lt; \sigma &lt; 2 $ and $ \frac{4}{N} &lt; \alpha &lt; \frac{4}{N-2} $. By using the concentration compactness principle and considering a local minimization problem, we prove that there exists a $ \gamma_0 &gt; 0 $ sufficiently small such that $ 0 &lt; \gamma &lt; \gamma_0 $ and for any $ a\in(0, a_0) $, there exist stable standing waves for the problem in the $ L^2 $-supercritical case. Our results are complement to the result of Li-Zhao in ^{[<xref ref-type="bibr" rid="b23">23</xref>]}.</p></abstract>