Multiplicity of positive solutions of a nonlinear Schrödinger equation

We consider the multiple existence of positive solutions of the following nonlinear Schrödinger equation: [inline-graphic not available: see fulltext] where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{p\in (1, {{N+2}\over{ N-2}})}}$\end{document} if N≥3 and p(1, ∞) if N=1,2, and a(x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well Ω := int a−1(0) consisting of k components \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\Omega_1, \ldots, \Omega_k}}$\end{document} and the first eigenvalues of −Δ+b(x) on Ωj under Dirichlet boundary condition are positive for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{j=1,2,\ldots,k}}$\end{document}. Under these conditions we show that (PMλ) has at least 2k−1 positive solutions for large λ. More precisely we show that for any given non-empty subset \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{J\subset\{1,2,\ldots k\}}}$\end{document}, (Pλ) has a positive solutions uλ(x) for large λ. In addition for any sequence λn→∞ we can extract a subsequence λni along which uλni converges strongly in H1(RN). Moreover the limit function u(x)=limi→∞uλni satisfies (i) For jJ the restriction u|Ωj of u(x) to Ωj is a least energy solution of −Δv+b(x)v=vp in Ωj and v=0 on ∂Ωj. (ii) u(x)=0 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{x\in {\bf R}^N\setminus(\bigcup_{{j\in J}} \Omega_j)}}$\end{document}.