Multiplicity of positive solutions of a nonlinear Schrödinger equation

Abstract.

We consider the multiple existence of positive solutions of the following nonlinear Schrödinger 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 ${{x\in {\bf R}^N\setminus(\bigcup_{{j\in J}} \Omega_j)}}$.