Acta mathematica scientia, Series B >
BOUND STATES FOR A STATIONARY NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH SIGN-CHANGING POTENTIAL IN R3
Received date: 2009-04-09
Online published: 2009-07-20
Supported by
This work was supported by NSFC (10631030) and CAS-KJCX3-SYW-S03.
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.
JIANG Yong-Sheng , ZHOU Huan-Song . BOUND STATES FOR A STATIONARY NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH SIGN-CHANGING POTENTIAL IN R3[J]. Acta mathematica scientia, Series B, 2009 , 29(4) : 1095 -1104 . DOI: 10.1016/S0252-9602(09)60088-6
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