Existence and concentration of ground state solutions for singularly perturbed nonlocal elliptic problems

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• 作者：Dengfeng Lü
• 关键词：Kirchhoff ; type equation ; Hartree ; type nonlinearity ; Variational method ; Ground state solution ; Concentration phenomena
• 刊名：Monatshefte für Mathematik
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：182
• 期：2
• 页码：335-358
• 全文大小：
• 刊物主题：Mathematics, general;
• 出版者：Springer Vienna
• ISSN：1436-5081
• 卷排序：182

文摘

We consider the following singularly perturbed nonlocal elliptic problem $$\begin{aligned} -\left( \varepsilon ^{2}a+\varepsilon b\displaystyle \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx\right) \Delta u+V(x)u=\displaystyle \varepsilon ^{\alpha -3}(W_{\alpha }(x)*|u|^{p})|u|^{p-2}u, \quad x\in \mathbb {R}^{3}, \end{aligned}$$where \(\varepsilon >0\) is a parameter, \(a>0,b\ge 0\) are constants, \(\alpha \in (0,3)\), \(p\in [2, 6-\alpha )\), \(W_{\alpha }(x)\) is a convolution kernel and V(x) is an external potential satisfying some conditions. By using variational methods, we establish the existence and concentration of positive ground state solutions for the above equation.KeywordsKirchhoff-type equationHartree-type nonlinearity Variational methodGround state solutionConcentration phenomenaCommunicated by A. Jüngel.