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

doi:10.1016/S0252-9602(09)60088-6

Abstract

Abstract:

We study the following Schr¨odinger-Poisson system

(P_λ) −Δu + V (x)u + λφ(x)u = Q(x)u^p, 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 (P_λ) with p ∈ (2, 5) has always a positive radial solution, but (P_λ) 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 (P_λ) 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 u_λ of (P_λ) converges, along a subsequence, to a solution of (P₀) in H¹as λ → 0.

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

CLC Number:

35J60

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.

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

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