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

Original · EN

We study the following Schrödinger-Poisson system (Pλ)ll -Δu + V(x)u+λϕ(x) u =Q(x)uᵖ, x∈ R³ -Δϕ= u², |ₓ|→ ₊∞ϕ(x)=0, u>0, where λ0 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 RuizD-JFA 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 λ1/4. 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.