Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential

Abstract

We are concerned with the existence of single- and multi-bump solutions of the equation mathematical equation represented here p > 2, and p < 2N/N-2 if N ≥ 3. We require that a ≥ 0 is in L ∞loc(ℝ N) and has a bounded potential well Ω, i.e. a(x) = 0 for x ∈ Ω and a(x) > 0 for x ∈ ℝ N \ Ω . Unlike most other papers on this problem we allow that a 0 ∈ L ∞(ℝ N) changes sign. Using variational methods we prove the existence of multibump solutions u λ which localize, as λ → ∞, near prescribed isolated open subsets Ω 1, ⋯, Ω k ⊂ Ω. The operator L 0 := -Δ+a 0 may have negative eigenvalues in Ω j, each bump of u λ may be sign-changing.