本文与文献[17]研究$\u211d^N$中分数阶Hardy--Hénon方程$(-Δ)^s u = |x|^\ell |u|^{p-1} u$的稳定（相应地，外部稳定）弱解的存在性与不存在性，其中$0 < s < 1$，$\ell > -2s$，$p>1$，$N \geq 1$且$N > 2s$。本文证明了Joseph--Lundgren次临界情形下的不存在性结论。
英文摘要：
This paper and [17] treat the existence and nonexistence of stable (resp. outside stable) weak solutions to a fractional Hardy--Hénon equation $(-Δ)^s u = |x|^\ell |u|^{p-1} u$ in $\mathbb{R}^N$ where $0 < s < 1$, $\ell > -2s$, $p>1$, $N \geq 1$ and $N > 2s$. In this paper, the nonexistence part is proved for the Joseph--Lundgren subcritical case.
In this paper we deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.
In this paper we study nonexistence of non-negative distributional supersolutions for a class of semilinear elliptic equations involving inverse-square potentials.
We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$ to the following fractional Hardy-Hénon system $$ \left\{\begin{aligned} &(-Δ)^{s_1}u(x)=|x|^αv^p(x),\ \ x\in\mathbb{R}_+^n, \\&(-Δ)^{s_2}v(x)=|x|^βu^q(x),\ \ x\in\mathbb{R}_+^n, \\&u(x)\geq 0,\ \ v(x)\geq 0,\ \ x\in\mathbb{R}_+^n, \end{aligned}\right. $$ 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 $α,β$ via the methods of moving spheres and moving planes. Particularly, for the case $s_1=s_2$, one …
In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-Δ)^\fracθ{2} u=|x|^{-γ} u^p ,\quad x\in{\bf R}^N, t>0, \qquad u(0)=μ\quad \mbox{in} \quad {\bf R}^N, \] where $N\ge 1$, $0<θ\le2$, $p>1$, $γ>0$ and $μ$ is a nonnegative Radon measure on ${\bf R}^N$. Using these conditions, we attempt to identify the optimal strength of the singularity of $μ$ for the existence of solutions to this problem.
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We present the best constant and the existence of extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in $\mathbb{R}^N$. We also discuss the connection of the related functional spaces and as a result we obtain some Caffarelli - Kohn - Nirenberg inequalities. Our starting point is the existence of a minimizer for the Bliss' inequality and the indirect dependence of the Hardy inequality at the origin.
In this note we present the Pohozaev identity for the fractional Laplacian. As a consequence of this identity, we prove the nonexistence of nontrivial bounded solutions to semilinear problems with supercritical nonlinearities in star-shaped domains.
In this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the Hardy potential.
In this paper, we give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains. Our methods are self-contained and new. The main ideas and key ingredients will be discussed in the next section after Theorem 1.1 for punctured domains, and after Theorem 1.4 for exterior domains. We will also explain why all the previous methods and techniques do not apply to our general setting. Let us give here a foretaste of our line of attack: Based on the imbalance between the Hard…
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In this paper, we are interested to analyze a nonlocal nonlinear parabolic equation with fractional Laplacian. We show that there are no nontrivial global weak solutions using the test function method.
In this paper, we consider the Cauchy problem in $\mathbb{R}^N$, $N\geq1$, for semi-linear Schrödinger equations with space-time fractional derivatives. We discuss the nonexistence of global $L^1$ or $L^2$ weak solutions in the subcritical and critical cases under some conditions on the initial data and the nonlinear term. Furthermore, the nonexistence of local $L^1$ or $L^2$ weak solutions in the supercritical case are studied.
This paper is devoted to study the following Choquard equation \begin{eqnarray*}\left\{ \begin{array}{lll} (-\triangle)^{α/2}u=(|x|^{β-n}\ast u^p)u^{p-1},~~~&x\in R^n, u\geq0,\,\,&x\in R^n, \end{array} \right. \end{eqnarray*} where $0<α,β<2$, $1\leq p<\infty$, and $n\geq2$. Using a direct method of moving planes, we prove the symmetry and nonexistence of positive solutions in the critical and subcritical case respectively.
This paper is devoted to study the nonexistence results of positive solutions for the following fractional H$\acute{e}$non system \begin{eqnarray*}\left\{ \begin{array}{lll} &(-\triangle)^{α/2}u=|x|^av^p,~~~&x\in R^n, &(-\triangle)^{α/2}v=|x|^bu^q,~~~ &x\in R^n, &u\geq0, v\geq 0, \end{array} \right. \end{eqnarray*} where $0<α<2$, $0<p,q<\infty$, $a$, $b$ $\geq0$, $n\geq2$. Using a direct method of moving planes, we prove non-existence of positive solution in the subcritical case.
We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-Δ)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first prove that all positive bounded solutions are monotone increasing along the $x_1$ direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains. To this end, we introduce seve…
We study a class of fractional elliptic problems of the form $\Ds u= f(u)$ in the half space $\R^N_+:=\{x \in \R^N\::\: x_1>0\}$ with the complementary Dirichlet condition $u \equiv 0$ in $\R^N \setminus \R^N_+$. Under mild assumptions on the nonlinearity $f$, we show that bounded positive solutions are increasing in $x_1$. For the special case $f(u)=u^q$, we deduce nonexistence of positive bounded solutions in the case where $q \ge 1$ and $q<\frac{N-1+2s}{N-1-2s}$ if $N \ge 1+2s$. We do not require integrability assumptions on the solutions we study.
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Existence and nonexistence of solutions for fractional elliptic equations arising from closed MEMS model
Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Jiangxi Normal University（江西师范大学）；Jiangsu University（江苏大学）
In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, \label{0.1} {& Δu + λ\frac{u^{2^*(s_1)-1}}{|x|^{s_1}} + \frac{u^{2^*(s_2)-1}}{|x|^{s_2}} =0 \text{in} Ω, & u=0 \qquad \text{on} Ω, where $0 \le s_2 < s_1 \le 2$, $0 \ne λ\in \Bbb R$ and $0 \in \partial Ω$. The existence (or nonexistence) for least-energy solutions has been extensively studied when $s_1=0$ or $s_2=0$. In this paper, we prove that if $0< s_2 < s_1 <2$ and the mean curvature of $\partial Ω$ at 0 $H(0)<0$, then \eqref{0.1} has a least-energy solution. Therefore, this paper has completed th…
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