Existence and Multiplicity of Solutions for Semilinear Elliptic Systems with Periodic Potential View Full Text


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Article Info

DATE

2017-09-20

AUTHORS

Guofeng Che, Haibo Chen, Liu Yang

ABSTRACT

In this paper, we consider the following semilinear elliptic systems: -Δu+V(x)u=Fu(x,u,v),inRN,-Δv+V(x)v=Fv(x,u,v),inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+V(x)u=F_{u}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$\end{document}where V:RN→R,Fu(x,u,v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V:\mathbb {R}^{N}\rightarrow \mathbb {R},~F_{u}(x,u,v)$$\end{document} and Fv(x,u,v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{v}(x,u,v)$$\end{document} are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of -▵+V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\triangle +V$$\end{document}. Under appropriate assumptions on Fu(x,u,v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{u}(x, u, v)$$\end{document} and Fv(x,u,v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{v}(x, u, v)$$\end{document}, we prove the above system has a ground-state solution by using the Nehari-type technique in a strongly indefinite setting. Furthermore, the existence of infinitely many geometrically distinct solutions is obtained via variational methods. Recent results from the literature are improved and extended. More... »

PAGES

1329-1348

References to SciGraph publications

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