Existence and multiplicity of solutions for a class of elliptic problem with critical growth

被引:0
作者
Claudianor O. Alves
Luciano M. Barros
机构
[1] Unidade Acadêmica de Matemática,Universidade Federal de Campina Grande
[2] Unidade Acadêmica de Física e Matemática,Universidade Federal de Campina Grande
来源
Monatshefte für Mathematik | 2018年 / 187卷
关键词
Positive solutions; Critical exponents; Variational methods; 35B09; 35B33; 35A15;
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摘要
In this paper we show the existence and multiplicity of positive solutions for a class of elliptic problem of the type -Δu+λV(x)u=μuq-1+u2∗-1,inRN,(P)λ,μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\,\Delta u+\lambda V(x)u=\mu u^{q-1}+u^{2^*-1},\quad \text{ in } \quad \mathbb {R}^N, \qquad {(P)_{\lambda , \mu }} \end{aligned}$$\end{document}where λ,μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda , \mu >0$$\end{document}, q∈(2,2∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \in (2,2^*)$$\end{document} and V:RN→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V:\mathbb {R}^N \rightarrow \mathbb {R}$$\end{document} is a continuous function verifying some conditions. By using variational methods, we have proved that the above problem has at least cat(int(V-1)({0}))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$cat(int(V^{-1}) (\{0\}))$$\end{document} of positive solutions if λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is large and μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is small.
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页码:195 / 215
页数:20
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