Stationary nonlinear Schrödinger equations in \(\mathbb {R}^2\) with potentials vanishing at infinity

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• 作者：João Marcos do Ó ; Federica Sani…
• 关键词：Nonlinear Schrödinger equation ; Bound state ; Vanishing potentials ; Trudinger–Moser inequality ; Exponential growth
• 刊名：Annali di Matematica Pura ed Applicata (1923 -)
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：196
• 期：1
• 页码：363-393
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1618-1891
• 卷排序：196

文摘

We deal with a class of 2-D stationary nonlinear Schrödinger equations (NLS) involving potentials V and weights Q decaying to zero at infinity as \((1+|x|^\alpha )^{-1}\), \(\alpha \in (0,2)\), and \((1+|x|^\beta )^{-1}\), \(\beta \in (2, + \infty )\), respectively, and nonlinearities with exponential growth of the form \(\exp {\gamma _0 s^2}\) for some \(\gamma _0>0\). Working in weighted Sobolev spaces, we prove the existence of a bound state solution, i.e. a solution belonging to \(H^1({{\mathrm{\mathbb {R}}}}^2)\). Our approach is based on a weighted Trudinger–Moser-type inequality and the classical mountain pass theorem.