数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (4): 1095-1104.doi: 10.1016/S0252-9602(09)60088-6

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BOUND STATES FOR A STATIONARY NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH SIGN-CHANGING POTENTIAL IN R3

蒋永生,周焕松   

  1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences P.O.Box 71010, Wuhan 430071, China
  • 收稿日期:2009-04-09 出版日期:2009-07-20 发布日期:2009-07-20
  • 基金资助:

    This work was supported by NSFC (10631030) and CAS-KJCX3-SYW-S03.

BOUND STATES FOR A STATIONARY NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH SIGN-CHANGING POTENTIAL IN R3

 JIANG Yong-Sheng, ZHOU Huan-Song   

  1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences P.O.Box 71010, Wuhan 430071, China
  • Received:2009-04-09 Online:2009-07-20 Published:2009-07-20
  • Supported by:

    This work was supported by NSFC (10631030) and CAS-KJCX3-SYW-S03.

摘要:

We study the following Schr¨odinger-Poisson system

 (Pλ)    −Δu + V (x)u + λφ(x)u = Q(x)up, x ∈R3,
             −Δφ = u2, lim|x|→+∞ φ (x) = 0, u > 0,

where λ > 0 is a parameter, 1 < p < +1, V (x) and Q(x) are sign-changing or non-positive functions in L(R3). When V (x) ≡ Q(x) ≡ 1, D. Ruiz [19] proved that (Pλ) with p ∈ (2, 5) has always a positive radial solution, but (Pλ) with p ∈ (1, 2] has solution only if λ > 0 small enough and no any nontrivial solution if λ ≥1/4 . By using sub-supersolution method, we prove that there exists λ0 > 0 such that (Pλ) with p ∈ (1,+∞) has always a bound
state (H1(R3) solution) for λ ∈ [0, λ0) and certain functions V (x) and Q(x) in L(R3). Moreover, for every λ ∈ [0, λ0), the solutions uλ of (Pλ) converges, along a subsequence, to a solution of (P0) in H1 as λ → 0.

关键词: Schrödinger-Poisson system, sub-supersolutions, supercritical Sobolev exponent, sign-changing potential, bound state

Abstract:

We study the following Schr¨odinger-Poisson system

 (Pλ)    −Δu + V (x)u + λφ(x)u = Q(x)up, x ∈R3,
             −Δφ = u2, lim|x|→+∞ φ (x) = 0, u > 0,

where λ > 0 is a parameter, 1 < p < +1, V (x) and Q(x) are sign-changing or non-positive functions in L(R3). When V (x) ≡ Q(x) ≡ 1, D. Ruiz [19] proved that (Pλ) with p ∈ (2, 5) has always a positive radial solution, but (Pλ) with p ∈ (1, 2] has solution only if λ > 0 small enough and no any nontrivial solution if λ ≥1/4 . By using sub-supersolution method, we prove that there exists λ0 > 0 such that (Pλ) with p ∈ (1,+∞) has always a bound
state (H1(R3) solution) for λ ∈ [0, λ0) and certain functions V (x) and Q(x) in L(R3). Moreover, for every λ ∈ [0, λ0), the solutions uλ of (Pλ) converges, along a subsequence, to a solution of (P0) in H1 as λ → 0.

Key words: Schrödinger-Poisson system, sub-supersolutions, supercritical Sobolev exponent, sign-changing potential, bound state

中图分类号: 

  • 35J60