Laplace’s equation with concave and convex boundary nonlinearities on an exterior region

doi:10.1186/s13661-019-1163-7

	Laplace’s equation with concave and convex boundary nonlinearities on an exterior region
	Mao,Jinxiu; Zhao,Zengqin; Qian,Aixia
刊名	Boundary Value Problems
	2019-03-13
卷号	2019 期号:1
关键词	Exterior regions Laplace operator Concave and convex mixed nonlinear boundary conditions Fountain theorems Steklov eigenvalue problems 35J20 35J65 46E22 49R99
ISSN号	1687-2770
DOI	10.1186/s13661-019-1163-7
英文摘要	AbstractThis paper studies Laplace’s equation ?Δu=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$-\Delta u=0$\end{document} in an exterior region U?RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$U\varsubsetneq {\mathbb{R}}^{N}$\end{document}, when N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq 3$\end{document}, subject to the nonlinear boundary condition ?u?ν=λ\|u\|q?2u+μ\|u\|p?2u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{\partial u}{\partial \nu }=\lambda \vert u \vert ^{q-2}u+\mu \vert u \vert ^{p-2}u$\end{document} on ?U with 10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda >0$\end{document} and μ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu \in \mathbb{R}$\end{document} arbitrary, then there exists a sequence {uk}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{u_{k} \}$\end{document} of solutions with negative energy converging to 0 as k→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k\to \infty $\end{document}; on the other hand, when λ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda \in \mathbb{R}$\end{document} and μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu >0$\end{document} arbitrary, then there exists a sequence {u?k}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{\tilde{u}_{k} \}$\end{document} of solutions with positive and unbounded energy. Also, associated with the p-Laplacian equation ?Δpu=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$-\Delta _{p} u=0$\end{document}, the exterior p-harmonic Steklov eigenvalue problems are described.
语种	英语
出版者	Springer International Publishing
WOS记录号	BMC:10.1186/S13661-019-1163-7
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/32524]
专题	中国科学院数学与系统科学研究院
通讯作者	Mao,Jinxiu
作者单位
推荐引用方式 GB/T 7714	Mao,Jinxiu,Zhao,Zengqin,Qian,Aixia. Laplace’s equation with concave and convex boundary nonlinearities on an exterior region[J]. Boundary Value Problems,2019,2019(1).
APA	Mao,Jinxiu,Zhao,Zengqin,&Qian,Aixia.(2019).Laplace’s equation with concave and convex boundary nonlinearities on an exterior region.Boundary Value Problems,2019(1).
MLA	Mao,Jinxiu,et al."Laplace’s equation with concave and convex boundary nonlinearities on an exterior region".Boundary Value Problems 2019.1(2019).