Bound states for a stationary nonlinear Schrödinger-Poisson system with sign-changing potential in ℝ3

We study the following Schrödinger-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 < +∞, 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 ≥ ¼. By using sub-supersolution method, we prove that there exists λ0 > 0 such that (Pλ) with p ∈ (l,+∞) 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.