Multiplicity of positive solutions of a nonlinear Schrödinger equation

We consider the multiple existence of positive solutions of the following nonlinear Schrödinger equation: -Δu + (λa(x) + b(x))u = u ^p, u > 0 in R^N, where p ∈ (1, N+2/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 Ω₁, ... , Ω_k and the first eigenvalues of -Δ + b(x) on Ω _j under Dirichlet boundary condition are positive for all j = 1,2, ..., k. Under these conditions we show that (P_λ) has at least 2^k - 1 positive solutions for large λ. More precisely we show that for any given non-empty subset J ⊂ {1, 2,...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_ni; along which u_λni converges strongly in H¹ (R^N). Moreover the limit function u(x) = lim_{i → ∞} u_λni satisfies (i) For j ∈ J the restriction u |_Ωj of u(x) to Ω_j is a least energy solution of -Δu + b(x)v = u^p in Ω _j and u = 0 on ∂Ω_j. (ii) u(x) = 0 for X ∈ R^N \ (∪jεJ Ω_j).