7 results
On Liouville theorems of a Hartree–Poisson system
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- Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 4 / November 2023
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- 26 October 2023, pp. 1154-1178
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In this paper, we are concerned with the non-existence of positive solutions of a Hartree–Poisson system:
where
\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*}
$n \geq3$ 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.
Nonexistence of anti-symmetric solutions for fractional Hardy–Hénon system
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- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 154 / Issue 3 / June 2024
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- 15 May 2023, pp. 862-886
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- June 2024
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We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$
for the following fractional Hardy–Hénon system:\[ \left\{\begin{array}{@{}ll} (-\Delta)^{s_1}u(x)=|x|^\alpha v^p(x), & x\in\mathbb{R}_+^n, \\ (-\Delta)^{s_2}v(x)=|x|^\beta u^q(x), & x\in\mathbb{R}_+^n, \\ u(x)\geq 0, & v(x)\geq 0,\ x\in\mathbb{R}_+^n, \end{array}\right. \]where $0< s_1,s_2<1$
, $n>2\max \{s_1,s_2\}$
. Nonexistence of anti-symmetric solutions are obtained in some appropriate domains of $(p,q)$
under some corresponding assumptions of $\alpha,\beta$
via the methods of moving spheres and moving planes. Particularly, for the case $s_1=s_2$
, one of our results shows that one domain of $(p,q)$
, where nonexistence of anti-symmetric solutions with appropriate decay conditions at infinity hold true, locates at above the fractional Sobolev's hyperbola under appropriate condition of $\alpha, \beta$
.
5 - Fundamentals of Statistical Mechanics
- from Part II - Statistical Molecular Dynamics
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- Introduction to Computational Nanomechanics
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- 12 October 2022
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- 08 December 2022, pp 105-157
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STABLE SOLUTIONS TO THE STATIC CHOQUARD EQUATION
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- Bulletin of the Australian Mathematical Society / Volume 102 / Issue 3 / December 2020
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- 10 June 2020, pp. 471-478
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- December 2020
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This paper is concerned with the static Choquard equation
where
$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u=\bigg(\frac{1}{|x|^{N-\unicode[STIX]{x1D6FC}}}\ast |u|^{p}\bigg)|u|^{p-2}u\quad \text{in }\mathbb{R}^{N},\end{eqnarray}$$
$N,p>2$ and 
$\max \{0,N-4\}<\unicode[STIX]{x1D6FC}<N$. We prove that if 
$u\in C^{1}(\mathbb{R}^{N})$ is a stable weak solution of the equation, then 
$u\equiv 0$. This phenomenon is quite different from that of the local Lane–Emden equation, where such a result only holds for low exponents in high dimensions. Our result is the first Liouville theorem for Choquard-type equations with supercritical exponents and 
$\unicode[STIX]{x1D6FC}\neq 2$.
On critical exponents of a k-Hessian equation in the whole space
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- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 149 / Issue 6 / December 2019
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- 18 January 2019, pp. 1555-1575
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- December 2019
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In this paper, we study negative classical solutions and stable solutions of the following k-Hessian equation
with radial structure, where n ⩾ 3, 1 < k < n/2 and p > 1. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to research k-Hessian equations without radial structure.
$$F_k(D^2V) = (-V)^p\quad {\rm in}\;\; R^n$$
The Lane–Emden equation in strips
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- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 148 / Issue 1 / February 2018
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- 20 November 2017, pp. 51-62
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- February 2018
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Gradient estimates for semilinear elliptic systems and other related results
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- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 145 / Issue 6 / December 2015
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- 22 October 2015, pp. 1313-1330
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- December 2015
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