4 results
Liouville theorems for the sub-linear Lane–Emden equation on the half space
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- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
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- 18 November 2024, pp. 1-13
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In this article, we study the following Dirichlet problem to the sub-linear Lane–Emden equation
\begin{equation*}\left\{\begin{array}{ll}-\Delta u=u^{p},\quad u(x)\geq0,\quad x\in\mathbb{R}^n_+, \\u(x)\equiv0,\quad x\in\partial\mathbb{R}^n_+,\end{array}\right.\end{equation*}where
$n\geq3$, 
$0 \lt p\leq1$. By establishing an equivalent integral equation, we give a lower bound of the Kelvin transformation 
$\bar{u}$. Then, by constructing a new comparison function, we apply the maximum principle based on comparisons and the method of moving planes to obtain that u only depends on xn. Based on this, we prove the non-existence of non-negative solutions.
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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- Journal:
- 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 $\alpha, \beta$
.
A NOTE ON RADIAL SYMMETRY FOR AN INTEGRAL EQUATION OF WOLFF TYPE
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- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 2 / October 2019
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- 20 February 2019, pp. 323-327
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- October 2019
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We prove that positive solutions of an integral equation of Wolff type are radially symmetric and decreasing about some point in
$R^{n}$. The hypotheses allow a wider range of exponents and are easier to apply than those in previous work.
