On a class of nonlinear Schrödinger-Poisson systems involving a nonradial charge density

2018/05/02 by Mercuri, Carlo, Tyler, Teresa Megan
#35B65 #35J20 #35J60 #35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.00964

Abstract

In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schrödinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schrödinger-Poisson system \- Δu+ u + ρ(x) ϕu = |u|^p-1 u, · amp;x∈ \mathbb R³, -Δϕ=ρ(x) u², · amp; x∈ \mathbb R³, . under different assumptions on ρ: \mathbb R³→ \mathbb R₊ at infinity. Our results cover the range p∈(2,3) where the lack of compactness phenomena may be due to the combined effect of the invariance by translations of a `limiting problem' at infinity and of the possible unboundedness of the Palais-Smale sequences. Moreover, we find necessary conditions for concentration at points to occur for solutions to the singularly perturbed problem \- ε²Δu+ u + ρ(x) ϕu = |u|^p-1 u, · amp;x∈ \mathbb R³, -Δϕ=ρ(x) u², · amp; x∈ \mathbb R³, . in various functional settings which are suitable for both variational and perturbation methods.

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