On the radiality of constrained minimizers to the Schrödinger–Poisson–Slater energy

Abstract

We study the radial symmetry\ud of minimizers to the Schr\"odinger-Poisson-Slater\ud (S-P-S) energy:\ud $$\inf_{\substack{u\in H^1(\R^3)\\\ud \|u\|_{L^2(\R^3)}=\rho}}\ud \frac 12 \int_{\R^3} |\nabla u|^2 + \frac 14\ud \int_{\R^3} \int_{\R^3} \frac{|u(x)|^2|u(y)|^2}{|x-y|}dxdy\ud - \frac 1p \int_{\R^3} |u|^p dx$$\ud provided that $2<p<3$ and $\rho$ is small. The main result shows that minimizers are radially symmetric modulo suitable translation

Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations

“…However, when we work in all H 1 (R 3 ) it is still not known if ground states, or at least one of them, are radially symmetric. In that direction, we are only aware of the result of Georgiev, Prinari and Visciglia [16] which gives a positive answer when p ∈ (2, 3) and for c > 0 sufficiently small. In this range, the critical point is found as a minimizer of F (u) on S(c).…”

On the existence of ground states of an equation of Schrödinger–Poisson–Slater type

“…We also note that there are many results on other Schrödinger-Poisson systems in literatures; see [1,4,5,7,11] and many others. This paper is concerned with the variational problem of (4).…”

Existence of PrescribedL2-Norm Solutions for a Class of Schrödinger-Poisson Equation

“…If, in addition, the condition (8) holds, then one can show that dichotomy does not occur; that is, ∈ . Furthermore, if (20) and (21) are also fulfilled, then { } strongly converges to in 1 (R 3 ). Finally we recall the following results obtained in [17,18].…”

“…(c) As we have anticipated, the existence of minimizers for 2 is related to the existence and stability of the standing wave solutions to (2). For the existence of stable standing wave solutions to (2), we refer to [4,14,17,18,20,21] and the references therein.…”