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

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 systemunder different assumptions on ρ : R 3 → 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 problemin various functional settings which are suitable for both variational and perturbation methods. MSC: 35J20, 35B65, 35J60, 35Q55

Existence of Positive Ground State Solution for the Nonlinear Schrödinger-Poisson System with Potentials

“…Let ≥ 0 be fixed. The first inequality of (38) is a straight consequence of (31). In order to show the second inequality we should construct a sequence { } ⊂ N and lim Φ ( ) = ∞ .…”

“…More recently, many contributions to (1) have also been given looking at cases in which no symmetry assumptions are given on the coefficients appearing in (1); one can refer to the papers [22][23][24]. Furthermore, for more results on the existence of positive solutions, ground and bound states, one can see [18,19,21,[25][26][27][28][29][30][31][32][33] and references therein. Nearly, the paper [34] proves the existence of bound state solution of (1) under some decay condition on , , and .…”

Bound state nodal solutions for the non-autonomous Schrödinger–Poisson system in R3

“…In recent years, there has been much attention to SP systems like system (SP λ ) on the existence of positive solutions, ground states, radial solutions and semiclassical states. We refer the reader to [3,4,12,13,14,21,28,29,32,34,35,36,40]. More precisely, Ruiz [32] studied the autonomous SP system…”

Positive bound state solutions for the nonlinear Schrödinger–Poisson systems with potentials

“…Cerami and Molle [6] obtained the existence of bound state, finite energy solution of (1.3) under suitable assumptions on the decay rate of the coefficients A, B, b. In [17], Mercuri and Tyler proved the existence of mountain-pass solutions and least energy solutions to the nonlinear Schrödinger-Poisson system (1.3) with A(y) = b(y) = 1 and p ∈ (2, 5) under different assumptions on B : R 3 → R + at infinity. Furthermore, they also studied the singularly perturbed problem and found necessary conditions for concentration at points to occur for solutions to the singularly perturbed problem in various functional settings.…”