Multiplicity of positive solutions of a nonlinear Schr�dinger equation

Abstract

We consider the multiple existence of positive solutions of the following nonlinear Schrodinger equation: where ${{p\in (1, {{N+2}\over{ N-2}})}}$ 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 ${{\Omega_1, \ldots, \Omega_k}}$ and the first eigenvalues of −Δ+b(x) on Ω j under Dirichlet boundary condition are positive for all ${{j=1,2,\ldots,k}}$ . Under these conditions we show that (PM λ) has at least 2 k −1 positive solutions for large λ. More precisely we show that for any given non-empty subset ${{J\subset\{1,2,\ldots k\}}}$ , (P λ ) has a positive solutions u λ (x) for large λ. In addition for any sequence λ n →∞ we can extract a subsequence λ n i along which u λni converges strongly in H 1 (R N ). Moreover the limit function u(x)=lim i→∞ u λni satisfies (i) For jJ the restriction u| Ω j of u(x) to Ω j is a least energy solution of −Δv+b(x)v=v p in Ω j and v=0 on ∂Ω j . (ii) u(x)=0 for ${{x\in {\bf R}^N\setminus(\bigcup_{{j\in J}} \Omega_j)}}$ .