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An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well
  • Kuan-Hsiang Wang
Kuan-Hsiang Wang
National University of Kaohsiung
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Abstract

In this paper, we study an eigenvalue problem for Schrödinger-Poisson system with indefinite nonlinearity and potential well as follows: -Δu+μV(x)u+K(x)Φu = λf(x)u+g(x)|u|p-2u in R3, -ΔΦ=K(x)u^2 in R3, where 4≤p<6, the parameters μ, λ>0, V∈ C(R3) is a potential well, and the functions f ∈ L3/2(R3) and g ∈ L(R3) are allowed to be sign-changing. It is well known that such a system with the potential being positive constant has two positive solutions when lim|x|→∞g(x)=g<0, K=0 in the set {x ∈ R3 : g(x)=0} and λ>λ1(f) with near λ1(f), where λ1(f) is the first eigenvalue of -Δ+ id in H1(R3) (see e.g. Huang et al., J. Differential Equations 255, 2463 (2013)). The main purpose is to obtain the existence and multiplicity of positive solutions without the above assumptions for g and K. The results are obtained via variational method and steep potential. Furthermore, we also consider the concentration of solutions as μ→∞.

Peer review status:UNDER REVIEW

21 Aug 2020Submitted to Mathematical Methods in the Applied Sciences
24 Aug 2020Assigned to Editor
24 Aug 2020Submission Checks Completed
01 Sep 2020Reviewer(s) Assigned