Regularity, symmetry and asymptotic behaviour of solutions for some Stein–Weiss-type integral systems

Michael Melgaard, Minbo Yang and Xianmei Zhou

Vol. 317 (2022), No. 1, 153–186

DOI: 10.2140/pjm.2022.317.153

Abstract

We consider the positive solutions of some integral systems related to the static Hartree-type equations:

$$\{\begin{array}{c}\hfill \\ \hfill \\ u(x)={\int \; <!--nolimits-->}_{{ℝ}^N}\frac{u^{p-1}(y)v(y)}{|x-y{|}^{N-\tau }}dy\hfill & \text{ in}{ℝ}^N,\hfill \\ & \\ v(x)={\int \; <!--nolimits-->}_{{ℝ}^N}\frac{u^p(y)}{\left|x{|}^{\alpha }\right|x-y{|}^{\mu }|y{|}^{\beta }}dy\hfill & \text{ in}{ℝ}^N,\hfill \\ \hfill \\ \hfill \\ \hfill \end{array}$$

where $N\ge 3$, $p\ge 1$, $0<\mu ,\tau <N$, $\alpha ,\beta \ge 0$ and $0<\alpha +\beta +\mu \le N$. Firstly, assuming that the exponent $p$ belongs to some suitable interval depending on the parameters $\mu ,\tau ,\alpha ,\beta$, we are able to prove some nonexistence results for the positive solution. Secondly, we also establish some qualitative results for the integrable solution of the system like regularity, symmetry and asymptotic behaviour. As a corollary, we deduce the corresponding results for the equivalent weighted Hartree-type nonlocal equations. The results obtained in this paper generalise and complement the existing results in the literature.

Keywords

integral systems, weighted Hartree equation, Riesz potential, integrable solution, decay rate

Mathematical Subject Classification

Primary: 35J15

Secondary: 45G05

Milestones

Received: 17 June 2021

Accepted: 17 March 2022

Published: 19 June 2022