### Abstract

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 Ω_{1}, ... , 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 H^{1} (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}).

Original language | English |
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Pages (from-to) | 109-135 |

Number of pages | 27 |

Journal | Manuscripta Mathematica |

Volume | 112 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2003 Sep 1 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Manuscripta Mathematica*,

*112*(1), 109-135. https://doi.org/10.1007/s00229-003-0397-x