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

TL;DR

The paper analyzes standing waves for a two-component Hartree-type system with Riesz potential, establishing the existence of positive ground states and detailing their qualitative properties. The authors employ a variational framework on $E=H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$, leveraging a Nehari manifold and mountain-pass geometry, together with polarization to prove radial symmetry about a common center. They develop a comprehensive regularity and integrability theory, and prove sharp exponential or nonstandard decay rates at infinity that depend on the relative sizes of $p$ and $q$, including polynomial-exponential cases when $p<2$ or $q<2$, and identify critical lines $p+q=\frac{2(N+α)}{N}$ and $p+q=2^*_α$ that delineate existence. A Pohozaev-type argument yields a nonexistence region, underscoring the sharpness of the exponent constraints. Collectively, the results extend scalar Choquard-type insights to a coupled Hartree system, advancing understanding of nonlocal interactions in multi-component models.

Abstract

We consider systems of the form \[ \left\{ \begin{array}{l} -Δu + u = \frac{2p}{p+q}(I_α\ast |v|^{q})|u|^{p-2}u \ \ \textrm{ in } \mathbb{R}^N, \\ -Δv + v = \frac{2q}{p+q}(I_α\ast |u|^{p})|v|^{q-2}v \ \ \textrm{ in } \mathbb{R}^N, \end{array} \right. \] for $α\in (0, N)$, $\max\left\{\frac{2α}{N}, 1\right\} < p, q < 2^*$ and $\frac{2(N+α)}{N} < p+ q < 2^{*}_α$, where $I_α$ denotes the Riesz potential, \[ 2^* = \left\{ \begin{array}{l}\frac{2N}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2, \end{array}\right. \quad \text{and} \quad 2^*_α = \left\{ \begin{array}{l}\frac{2(N+α)}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2. \end{array} \right. \] 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 $p+q=\frac{2(N+α)}{N}$ and $ p+ q = 2^{*}_α$ are critical for the existence of solutions.

Figures (7)

• Figure 1: Region on the $(p,q)$-plane corresponding to our existence results (Theorem \ref{['t3']}) in comparison to those in the literature. Here $N=3$ and $\alpha=1$.
• Figure 2: Regions on the $(p,q)$-plane corresponding to cases (a)-(d) of Theorem \ref{['t10novo']}. In this picture, $N=3$, $\alpha=1.9$.
• Figure 3: Regions on the $(p,q)$-plane corresponding to Theorem \ref{['t3']}, Theorem \ref{['t4']} and open problems. Here $N=3$ and $\alpha=1.9$.
• Figure 4: Points $(s,t)$ as in \ref{['precisamos']} and \ref{['sermaior1']} with $(p,q)$ as in Theorem \ref{['t10novo']} (a).
• Figure 5: Points $(s,t)$ as in \ref{['precisamos']} and \ref{['sermaior1']} with $(p,q)$ as in Theorem \ref{['t10novo']} (b).

Theorems & Definitions (52)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Proposition 2.1
• Proposition 2.2