Standing waves for nonlinear Hartree type equations: existence and qualitative properties

TL;DR

This paper studies the existence and properties of standing wave solutions for a coupled nonlinear Hartree system, identifying critical parameter lines and establishing qualitative features like symmetry and decay.

Contribution

It introduces new existence results and qualitative analysis for a class of coupled Hartree equations with critical parameter thresholds.

Findings

Existence of ground state solutions under certain parameter ranges.

Solutions exhibit radial symmetry and specific decay at infinity.

Identification of critical lines for parameters where solutions cease to exist.

Abstract

We consider systems of the form \[ \left\{ \begin{array}{l} -\Delta u + u = \frac{2p}{p+q}(I_\alpha \ast |v|^{q})|u|^{p-2}u \ \ \textrm{ in } \mathbb{R}^N, \\ -\Delta v + v = \frac{2q}{p+q}(I_\alpha \ast |u|^{p})|v|^{q-2}v \ \ \textrm{ in } \mathbb{R}^N, \end{array} \right. \] for $\alpha\in (0, N)$, $\max\left\{\frac{2\alpha}{N}, 1\right\} < p, q < 2^*$ and $\frac{2(N+\alpha)}{N} < p+ q < 2^{*}_{\alpha}$, where $I_\alpha$ denotes the Riesz potential, \[ 2^* = \left\{ \begin{array}{l}\frac{2N}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2, \end{array}\right. \quad \text{and} \quad 2^*_{\alpha} = \left\{ \begin{array}{l}\frac{2(N+\alpha)}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2. \end{array} \right. \] This type of systems arises in the study of standing wave solutions for a certain approximation of the Hartree theory for a two-component attractive interaction. We prove existence and some qualitative properties for ground state solutions, such as definite sign for each component, radial symmetry and sharp asymptotic decay at infinity, and a regularity/integrability result for the (weak) solutions. Moreover, we show that the straight lines $p+q=\frac{2(N+\alpha)}{N}$ and $ p+ q = 2^{*}_{\alpha}$ are critical for the existence of solutions.