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

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 $\alpha \in (0,N)$, $\max \left\{\frac{2\alpha }{N},1\right\}<p,q<2^{\ast }$ and $\frac{2(N+\alpha )}{N}<p+q<2_{\alpha }^{\ast }$, where $I_{\alpha }$ denotes the Riesz potential, \[ 2^* = \left\{ \begin{array}{l}\frac{2N}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2, \end{array}\right. \quad \text{and} \quad 2^*_α = \left\{ \begin{array}{l}\frac{2(N+α)}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2. \end{array} \right. \] 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 $p+q=\frac{2(N+\alpha )}{N}$ and $p+q=2_{\alpha }^{\ast }$ are critical for the existence of solutions.

35 pages, 7 figures, few typos were fixed

References in corpus (1)

• Existence and stability of standing waves for coupled nonlinear Hartree type equations