Uniqueness of non-negative solutions to an integral equation of the Choquard type

Le, Phuong

doi:10.1080/00036811.2022.2101452

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2023

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“…For Equation (1.2), there are several results. First, Le [25] pointed out that the solution of integral Equation (1.3) can be classified when p is the critical exponent, and hence u is classical. Similar to the proof of Theorem 1.1 in [9], we know that the distributional solutions of Equation (1.2) satisfy the integral equation when .…”

Section: Introductionmentioning

confidence: 99%

On Liouville theorems of a Hartree–Poisson system

Li,

Lei

2023

Proceedings of the Edinburgh Mathematical Society

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In this paper, we are concerned with the non-existence of positive solutions of a Hartree–Poisson system: \begin{equation*} \left\{ \begin{aligned} &-\Delta u=\left(\frac{1}{|x|^{n-2}}\ast v^p\right)v^{p-1},\quad u \gt 0\ \text{in} \ \mathbb{R}^{n},\\ &-\Delta v=\left(\frac{1}{|x|^{n-2}}\ast u^q\right)u^{q-1},\quad v \gt 0\ \text{in} \ \mathbb{R}^{n}, \end{aligned} \right. \end{equation*} where $n \geq3$ and $\min\{p,q\} \gt 1$ . We prove that the system has no positive solution under a Serrin-type condition. In addition, the system has no positive radial classical solution in a Sobolev-type subcritical case. In addition, the system has no positive solution with some integrability in this Sobolev-type subcritical case. Finally, the relation between a Liouville theorem and the estimate of boundary blowing-up rates is given.

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