Mokhtar Kirane, Ahmad Z. Fino, Bashir Ahmad
Some results on nonexistence of nontrivial solutions to some time and space fractional differential evolution equations with transformed space argument are obtained via the nonlinear capacity method. The analysis is then used for a $2\times 2$ system of equations with transformed space arguments.
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Fractional Laplacian: Pohozaev identity and nonexistence results
本文给出了分数拉普拉斯算子的Pohozaev恒等式。作为这个恒等式的结果，我们证明了在星形区域中具有超临界非线性项的半线性问题的非平凡有界解的不存在性。
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.
Nonexistence of positive supersolutions for semilinear fractional elliptic equations in exterior domains
本文研究ℝⁿ（n≥1）外区域中非线性分数阶椭圆不等式正解的不存在性，在极弱非线性项假设下得到新结果，且该结论在全空间情形下也具新颖性。
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.
On weak solutions to a fractional Hardy--Hénon equation: Part I: Nonexistence
该文是分数阶Hardy--Hénon方程弱解研究的第一部分，证明了Joseph--Lundgren次临界情形下该方程稳定及外部稳定弱解的不存在性。
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…
Monotonicity and nonexistence results for some fractional elliptic problems in the half space
本文研究了半空间$\R^N_+:=\{x \in \R^N\::\: x_1>0\}$中的一类具有互补Dirichlet条件$u \equiv 0$ in $\R^N \setminus \R^N_+$形式的分式椭圆问题.在对非线性方程$f$的适当假设下，证明了在$x_1$中有界正解是递增的.对于$f(u)=u^q$的特殊情形，当$q \ge 1$和$q<\frac{N-1+2s}{N-1-2s}$满足$N \ge 1+2s$时，我们得到了正有界解的不存在性.我们不需要可积性假设的解决方案，我们研究。
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 …
For the following semilinear equation with Hilfer- Hadamard fractional derivative \begin{equation*} \mathcal{D}^{α_1,β}_{a^+} u-Δ\mathcal{D}^{α_2,β}_{a^+} u-Δu =\vert u\vert^p, \qquad t>a>0, \qquad x\inΩ, \end{equation*} where $Ω\subset \mathbb{R}^N$ $(N\geqslant 1)$, $p>1$, $0<α_{2}<α_{1}<1$ and $0<β<1$. $\mathcal{D}^{α_i,β}_{a^+}$ $(i=1,2)$ is the Hilfer- Hadamard fractional derivative of order $α_i$ and of type $β$, we establish the necessary conditions for the existence of global solutions.
Nonexistence results for a class of fractional elliptic boundary value problems
本文研究了一类形如$$ \Ds u= f(x,u) \quad \textrm{in} Øu=0\quad \textrm{in} \R^N \setminus Ø,$$的分数阶椭圆问题，其中$s\in(0,1)$ .证明了当$Ø$为星形且$f$为超临界时正解的不存在性.我们还得到了亚临界$f$在某些无界区域上的不存在性结果。该论点依赖于移动球体的方法应用到一个重新制定的问题，使用Caffarelli-西尔维斯特扩展\cite{CSilv}的解决方案，上述问题。由于解的边界正则性的缺乏，使用Pohozaev型恒等式的$s=1$情况下的标准方法不适用于$0<s<1$情况。
Existence and nonexistence of solutions for fractional elliptic equations arising from closed MEMS model
本文研究了源于封闭MEMS模型的分数椭圆方程解的存在性与不存在性，分析了边界衰减对解和拉紧电压的影响。
Existence, Nonexistence, and Symmetry of Positive Solutions for Fractional Laplacian Problems
本文研究含分数阶 Laplace 算子的椭圆与抛物问题，利用山路引理和移动平面法证明次临界正解的存在性及对称性或单调性，并给出超临界与负指数情形的不存在性。
Global existence and nonexistence analyses for a magnetic fractional pseudo-parabolic equation
本文研究磁分数阶Orlicz-Sobolev空间中伪抛物方程的初边值问题，利用嵌入定理、位势井理论和Galerkin方法证明了全局解的存在唯一性、衰减估计及有限时间爆破，并探讨了与平稳问题基态解的收敛关系。
Nonexistence results for nonlocal equations with critical and supercritical nonlinearities
本文证明了一类非局部算子为\[Lu(x)=\sum a_{ij}\partial_{ij}u+{\rm PV}\int_{\R^n}(u(x)-u(x+y))K(y)dy.\]的非线性问题非平凡有界解的不存在性。这些算子是对称Lévy过程的无穷小生成元。我们的结果适用于甚至核$K$满足$K(y)|y|^{n+σ}$是非减少沿射线从原点，对于一些$σ\in(0,2)$的情况下$a_{ij}\equiv0$和$σ=2$的情况下，$(a_{ij})$是一个正定对称矩阵。我们的不存在性结果关注的Dirichlet问题的$L$在星形域的临界和超临界非线性（其中的临界条件是有关$n$和$σ$）。我们还建立了不存在的有界解的半线性方程，涉及其他非局部算子，如高阶分数拉普拉斯算子$(-Δ)^s$（这里$s>1$）或分数$p$ -拉普拉斯算子。所有这些不存在的结果遵循一个一般的变分不等式的精神，一个经典的身份由Pucci和Serrin。
In this paper, we discuss a test function method to obtain nonexistence of global-in-time solutions for higher order evolution equations with fractional derivatives and a power nonlinearity, under a sign condition on the initial data. In order to deal with fractional powers of the Laplace operator, we introduce a suitable test function and a suitable class of weak solutions. The optimality of the nonexistence result provided is guaranteed by both scaling arguments and counterexamples. In particular, our manuscript provides the counterpart of nonexistence for several recent results of global ex…
Semilinear elliptic equations for the fractional Laplacian with Hardy potential
本文研究一类含Hardy位势的半线性分数阶椭圆型方程非负分布解的存在性和不存在性。
On fractional Choquard equations
研究了一类具有广义Choquard非线性项和分数阶扩散的非线性Schrodinger方程.我们得到了正则性、存在性、不存在性、对称性以及衰减性。
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…
We discuss on existence, nonexistence and uniqueness of positive viscosity solutions for Lane-Emden systems involving the fractional Laplacian on bounded domains. As a byproduct, we obtain the critical hyperbole associated to the these systems.
In this article, we investigate the existence, uniqueness, nonexistence, and regularity of weak solutions to the nonlinear fractional elliptic problem of type $(P)$ (see below) involving singular nonlinearity and singular weights in smooth bounded domain. We prove the existence of weak solution in $W_{loc}^{s,p}(Ω)$ via approximation method. Establishing a new comparison principle of independent interest, we prove the uniqueness of weak solution for $0 \leq δ< 1+s- \frac{1}{p}$ and furthermore the nonexistence of weak solution for $δ\geq sp.$ Moreover, by virtue of barrier arguments we study t…
We study integrability conditions for existence and nonexistence of a local-in-time integral solution of fractional semilinear heat equations with rather general growing nonlinearities in uniformly local $L^p$ spaces. Our main results about this matter consist of Theorems 1.4, 1.6, 5.1 and 5.3. We introduce a new supersolution which plays a crucial role. Our method does not rely on a change of variables, and hence it can be applied to a wide class of nonlocal parabolic equations. In particular, when the nonlinear term is $u^p$ or $e^u$, a local-in-time solution can be constructed in the critic…
Positive solutions to some asymptotically linear fractional Schrodinger equations
本文证明了RN中一类分数阶Schrodinger方程正解的存在性和不存在性，当f（u）在无穷远处关于u是渐近线性时，利用一种新的方法得到了正解的存在性.
Nonexistence of traveling wave solutions in the fractional Rosenau-Hyman equation via homotopy perturbation method
该研究应用同伦摄动法构造分数阶Rosenau-Hyman方程的级数解，发现空间非局域性阻止紧孤立子存在，二次型方程因时间分数阶导数阶数分岔导致波传播固定，三次型可能有限时间爆破，同时指出同伦摄动法存在发散情况需进一步研究。
Our purpose of this paper is to study the nonexistence of nonnegative very weak solutions of \begin{equation}\label{eq 0.1} \displaystyle (-Δ)^αu = u^p+ν\quad {\rm in}\quad Ω,\qquad\ u=g\quad {\rm in}\quad \mathbb{ R}^N\setminus Ω, \end{equation} where $α\in(0,1)$, $p>0$, $Ω$ is a unbounded $C^2$ domain in $\mathbb{ R}^N$ with $N>2α$, $g\in L^1(\mathbb{ R}^N\setminus Ω,\frac{dx}{1+|x|^{N+2α}})$ nonnegative and $ν$ is a nonnegative Radon measure. We obtain that\smallskip $(i)$ if $Ω\supseteq \left(\mathbb{R}^N\setminus \overline{B_{r_0}(0)}\right)$ for some $r_0>0$ and $p<\frac{N}{N-2α}$, then …
In this paper, we are concerned with stable solutions to the fractional elliptic equation $$ (-Δ)^s u=e^u\mbox{ in }\mathbb R^{N}, $$ where $(-Δ)^s$ is the fractional Laplacian with $0<s<1$. We establish the nonexistence of stable solutions provided that $N<10s$. This result is optimal when $s\uparrow 1$.
Liouville type results for the fractional truncated Laplacians in a half-space
本文针对半空间中含分数阶截断拉普拉斯算子和幂型非线性项的完全非线性积分方程，研究其粘性上解的存在性，以推导区分存在与不存在情形的阈值指数估计。
We deal with the higher-order fractional Laplacians by two methods: the integral method and the system method. The former depends on the integral equation equivalent to the differential equation. The latter works directly on the differential equations. We first derive nonexistence of positive solutions, often known as the Liouville type theorem, for the integral and differential equations. Then through an delicate iteration, we show symmetry for positive solutions.
Fractional nonlinear Schrödinger equations with singular potential in $\mathbf R^n$
本文研究具有\begin{equation*} (-Δ)^su=\fracλ{|x|^α}u+|u|^{p-1}u,\quad \mathbf R^n\setminus\{0\}, \end{equation*}形式奇异位势的非线性分数阶薛定谔方程，其中$s\in (0,1)$，$α>0$，$p\ge1$和$λ\in \mathbf R$ .利用Caffarelli-Silvestre延拓方法，我们得到了该方程解的存在性、不存在性、正则性和对称性.
In this paper, we prove the existence of unbounded sequence of eigenvalues for the fractional $p-$Laplacian with weight in $\mathbb{R}^N.$ We also show a nonexistence result when the weighthas positive integral. In addition, we show some qualitative properties of the first eigenfunction including a sharp decay estimate. Finally, we extend the decay result to the positive solutions of a Schrödinger type equation.
Let $0<α<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \begin{equation}(-\bigtriangleup)^{α/2} u(x)=f(u),\end{equation} on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. Applying a direct method of moving planes for the fractional Laplacian, we prove symmetry and nonexistence of positive solutions on $\mathbb{R}^n$ and $\mathbb{R}^n_+$ under mild conditions on $f$.
Singular solutions of fractional elliptic equations with absorption
研究了具有吸收$$ \left\{\arraycolsep=1pt \begin{array}{lll} (-Δ)^αu+|u|^{p-1}u=0,\quad & \rm{in}\quadΩ\setminus\{0\},\\[2mm] u=0,\quad & \rm{in}\quad \R^N\setminusΩ,\\[2mm] \lim_{x\to 0}u(x)=+\infty, \end{array} \right. $$的分数阶椭圆型方程的奇异解，其中$p>0$，$Ω$是$0\inΩ$的$\R^N\ (N\ge2)$的开、有界、光滑区域.我们分析了解的存在性、不存在性、唯一性和渐近性态。
In this paper, we prove sharp blow-up and global existence results for a time fractional diffusion-wave equation with a nonlinear memory term in a bounded domain, where the fractional derivative in time is taken in the sense of Caputo type. Moreover, we also give a result for nonexistence of global solutions to a wave equation with a nonlinear memory term in a bounded domain. The proof of blow-up results is based on the eigenfunction method and the asymptotic properties of solutions for an ordinary fractional differential inequality.
In this paper, we study the Pohozaev identity associated with a H$\acute{e}$non-Lane-Emden system involving the fractional Laplacian: \begin{equation} \left\{\begin{array}{ll} (-\triangle)^su=|x|^av^p,&x\inΩ, (-\triangle)^sv=|x|^bu^q,&x\inΩ, u=v=0,&x\in R^n\backslashΩ, \end{array} \right. \end{equation} in a star-shaped and bounded domain $Ω$ for $s\in(0,1)$. As an application of our identity, we deduce the nonexistence of positive solutions in the critical and supercritical cases.
In the present paper, we study an inhomogeneous pseudo-parabolic equation with nonlocal nonlinearity $$u_t-kΔu_t-Δu=I^γ_{0+}(|u|^{p})+ω(x),\,\ (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ where $p>1,\,k\geq 0$, $ω(x)\neq0$ and $I^γ_{0+}$ is the left Riemann-Liouville fractional integral of order $γ\in(0,1).$ Based on the test function method, we have proved the blow-up result for the critical case $γ=0,\,p=p_c$ for $N\geq3$, which answers an {\bf open question} posed in \cite{Zhou}, and in particular when $k=0$ it improves the result obtained in \cite{Bandle}. An interesting fact is that in the cas…
In this paper, we study the following critical system with fractional Laplacian: \begin{equation*} \begin{cases} (-Δ)^{s}u= μ_{1}|u|^{2^{\ast}-2}u+\frac{αγ}{2^{\ast}}|u|^{α-2}u|v|^β \ \ \ \text{in} \ \ \mathbb{R}^{n}, (-Δ)^{s}v= μ_{2}|v|^{2^{\ast}-2}v+\frac{βγ}{2^{\ast}}|u|^α|v|^{β-2}v\ \ \ \ \text{in} \ \ \mathbb{R}^{n}, u,v\in D_{s}(\mathbb{R}^{n}). \end{cases} \end{equation*} By using the Nehari\ manifold,\ under proper conditions, we establish the existence and nonexistence of positive least energy solution of the system.
Global branching for semilinear fractional Laplace with sublinear nonlinearity
本文研究次线性分数阶椭圆问题$(P_λ^s)$正解的存在性、不存在性和多重性，通过先验估计、上下解方法及经典连接定理证明至少存在两个不同正弱解，扩展了已有结果至非局部情形。
Self-generated interior blow-up solutions in fractional elliptic equation with absorption
本文研究了分数Laplacian算子$(E)$-$(-Δ)^α u(x)+|u|^{p-1}u(x)=0 in x\inΩ\setminus\mathcal{C}$在条件$u(x)=0$-$x\inΩ^c$和$\lim_{x\inΩ\setminus\mathcal{C}, x\to\mathcal{C}}u(x)=+\infty$下的正解问题，其中$p>1$和$Ω$是$\mathbb{R}^N$中的开有界$C^2$整环，$\mathcal{C}\subset Ω$是一个具有$N-1$多重维数的无边界紧致$C^2$流形，带有$α\in(0,1)$的算子$(-Δ)^α$是分数Laplacian算子.研究了问题$(E)$正解的存在性.此外，我们进一步分析了唯一性，渐近性和不存在性。
In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian \[ (-\De)^s u = \left( \int_{\Om}\frac{|u|^{2^*_{μ,s}}}{|x-y|^μ}\mathrm{d}y \right)|u|^{2^*_{μ,s}-2}u +\la u \; \text{in } \Om,\] where $\Om $ is a bounded domain in $\mathbb R^n$ with Lipschitz boundary, $\la $ is a real parameter, $s \in (0,1)$, $n >2s$ and $2^*_{μ,s}= (2n-μ)/(n-2s)$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, multiplicity, regularity and nonexistence results for solution of the above equation …
In this paper, we study the existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} (-Δ)^su+V(x)u+ϕu=|u|^{p-1}u, & \hbox{in $\mathbb{R}^3$,} (-Δ)^sϕ=u^2,& \hbox{in $\mathbb{R}^3$,} \end{array} \right. \end{equation*} where $2<p<2_s^{\ast}-1 = \frac{3+2s}{3-2s}$, $s\in(\frac{3}{4},1)$. Under certain assumptions on $V$, a nontrivial ground state solution $(u,ϕ)$ is established through using a monotonicity trick and global compactness Lemma. As its supplementary results, we prove some nonexistence results in the case …
Classification of solutions to a weighted singular fractional problem in the half space
研究带权奇异性分数问题在半空间中正解的分类，证明在特定条件下解具有一维对称性和单调性，并通过渐近s阶斜率完成分类，当参数超出范围时证明全局正解不存在。
Fractional elliptic reaction-diffusion systems with coupled gradient terms and different diffusion
本文研究带不同阶分数阶扩散与耦合梯度项的非局部椭圆系统非负解的存在性，通过加权Lebesgue空间下的新精细估计得到存在性结果，兼具理论意义。
A semilinear equation involving the fractional Laplacian in $\mathbb{R}^n$
本文考虑欧氏空间$\mathbb{R}^n$中含分数拉普拉斯算子的半线性方程：\begin{equation} (-Δ)^{α/2} u(x) = f(x_n) \,u^p(x), \quad x \in \mathbb{R}^n \label{n26} \end{equation}在次临界情形下的解，其中有$1<p<\frac{n+α}{n-α}$ .我们没有直接研究伪微分方程（\ref{n26}），而是首先在积分方程中寻找它的等价形式：\begin{equation} u(x)=\int_{\mathbb{R}^n}G_{\infty}(x,y)\,f(y_n)\, u^{p}(y)\,dy, \label{n27} \end{equation}其中$ G_{\infty}(x,y)$是与$\mathbb{R}^n$中的分数拉普拉斯算子相关的格林函数。利用\emph{method of moving planes in integral forms}，我们能够得到次临界情形下（\ref{n27}）正解的不存在性.因此，对于（\ref{n26}）也是同样的结论。
In this paper, we first establish a narrow region principle and a decay at infinity theorem to extend the direct method of moving planes for general fractional $p$-Laplacian systems. By virtue of this method, we can investigate the qualitative properties of the following Schrödinger system with fractional $p$-Laplacian \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} \left(-Δ\right)_{p}^{s}u+au^{p-1}& =&f(u,v), \\[0.05cm] \left(-Δ\right)_{p}^{t}v+bv^{p-1}& =&g(u,v), \end{array}\right. \end{equation*} where $0<s,\,t<1$ and $2<p<\infty$. We obtain the radial symmetry in the unit ball o…
Logistic diffusion equations governed by the superposition of operators of mixed fractional order
本文研究了由混合分数阶算子叠加驱动的对数扩散方程，在敌对环境中探讨种群生存与灭绝的条件及机制。
We study the existence/nonexistence and qualitative properties of the positive solutions to the problem \begin{align*} (-Δ)^s u -θ\frac{u}{|x|^{2s}}&=u^p - u^q \quad\text{in }\,\, \mathbb{R}^N,\quad u > 0 \quad\text{in }\,\, \mathbb{R}^N, \quad u \in \dot{H}^s(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N), \end{align*} where $s\in (0,1)$, $N>2s$, $q>p\geq{(N+2s)}/{(N-2s)}$, $θ\in(0, Λ_{N,s})$ and $Λ_{N,s}$ is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions we mean, both the radial symmetry that is obtained by using the moving plane method in a nonlo…
In this article we present a simple and unified probabilistic approach to prove nonexistence of positive super-solutions for systems of equations involving potential terms and the fractional Laplacian in an exterior domain. Such problems arise in the analysis of a priori estimates of solutions. The class of problems we consider in this article is quite general compared to the literature. The main ingredient for our proofs is the hitting time estimates for the symmetric $α$-stable process and probabilistic representation of the super-solutions.
In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition $u \to 0$ at infinity with respect to the spacial variables to a polynomial growth on $u$ by constructing auxiliary functions.Then we derive monotonicity for the solutions in a half space $\mathbb{R}_+^n \times \mathbb{R}$ and obtain some new …
In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass: \begin{equation*} \begin{cases} (-Δ)^{s}u=λu +μ|u|^{q-2}u+|u|^{2_{s}^{\ast}-2}u,&x\in\mathbb{R}^{N}, \int_{\mathbb{R}^{N}}u^{2}dx=a^{2},\\ \end{cases} \end{equation*} where $(-Δ)^{s}$ is the fractional Laplacian, $0<s<1$, $N>2s$, $2<q<2_{s}^{\ast}=2N/(N-2s)$ is a fractional critical Sobolev exponent, $a>0$, $μ\in \mathbb{R}$. By using Jeanjean's trick in \cite{Jeanjean}, and the standard method which can be found in \cite{Brezis} to overcome the lack of compactness, we firs…
On a sharp inequality of Adimurthi-Druet type and extremal functions
本文研究全空间分数维Adimurthi-Druet型尖锐不等式的极值函数存在性与不存在性，推广了已有的Trudinger-Moser型不等式，并利用Carleson-Chang两步策略与爆破分析证明了次临界和临界情形下极大化函数的存在性，同时给出特殊次临界情形的不存在性结果。
On some Critical Problems for the Fractional Laplacian Operator
我们研究了低阶扰动对下面涉及分数阶拉普拉斯算子的临界椭圆问题正解存在性的影响：（-Δ）^{α/2}u=λu^q+u^{ \frac{N+α}{N-α} }，\quad u>0 & \quad in Ω，u=0& \quad on \partial Ω，$ $ where $ Ω \subset \mathbb{R}^N $ is a smooth bounded domain, $ N \ge 1 $, $ λ>0 $, $ 0<q< \frac{N+α}{N-α} $, $ 0<α< \min \{N，2\} $. For suitable conditions on $ α $ depending on $ q $, we prove: In the case $ q<1 $, there exist at least two solutions for every $ 0<λ<Λ $ and some $ 0 $, at least one if $ λ=Λ $, no solution if $ λ>Λ $. For $ q=1 $ we show existence of at least one solution for $ 0<λ<λ_1 $ and nonexistence for $ λ \ge λ_1 $. When $ q>1 $ the existence …
An Lp-theory for fractional stationary Navier-Stokes equations
该研究在Lebesgue与Lorentz空间框架下，建立含分数阶拉普拉斯算子的三维定常Navier-Stokes方程的Lp理论，给出解存在/不存在的充分条件，得到解的逐点衰减速率、渐近轮廓与全局正则性，还得到Liouville型唯一性结果及经典方程的新正则性准则。
In this paper, we obtain nonexistence results of positive solutions, and also the existence of an unbounded sequence of solutions that changing sign for some critical problems involving conformally invariant operators on the standard unit sphere, and the fractional Laplacian operator in the Euclidean space. Our arguments are based on a reduction of the initial problem in the Euclidean space to an equivalent problem on the standard unit sphere and vice versa, what together to blow up arguments, a variant of Pohozaev's type identity, a refinement of regularity results for this type operators, an…
Kirchhoff-type equations involving the Fractional $(p,q)-$Laplacian
本文利用变分方法研究有界区域上Kirchhoff型分数阶$(p,q)$-Laplacian方程，通过直接极小化和山路定理证明至少两个弱解的存在性，并给出小参数$λ>0$时解不存在的结果。
On weak solutions to a fractional Hardy-Hénon equation: Part II: Existence
本文研究分数阶Hardy-Hénon方程的稳定弱解存在性，证明了Joseph-Lundgren临界/超临界指数下正径向稳定解族的存在性，还发现部分参数下Joseph-Lundgren临界指数具有$s=1$时不具备的多重性。
Fractional differential equations solved by using Mellin transform
本文提出用梅林变换求解多阶微分方程。它表明，有关的问题的转移的实部的论点，所产生的梅林积分操作的分数阶导数，可以克服。然后，可以找到任何涉及多阶分数阶导数（或积分）的分数微分方程的解。通过求解一组线性代数方程组，在梅林域中找到解，其逆变换给出了手头的分数微分方程的解。
This paper deals with the following fractional Choquard equation $$\varepsilon^{2s}(-Δ)^su +Vu=\varepsilon^{-α}(I_α*|u|^p)|u|^{p-2}u\ \ \ \mathrm{in}\ \mathbb{R}^N,$$ where $\varepsilon>0$ is a small parameter, $(-Δ)^s$ is the fractional Laplacian, $N>2s$, $s\in(0,1)$, $α\in\big((N-4s)_{+}, N\big)$, $p\in[2, \frac{N+α}{N-2s})$, $I_α$ is a Riesz potential, $V\in C\big(\mathbb{R}^N, [0, +\infty)\big)$ is an electric potential. Under some assumptions on the decay rate of $V$ and the corresponding range of $p$, we prove that the problem has a family of solutions $\{u_\varepsilon\}$ concentrating a…
We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical problem \begin{equation*} (-Δ)^su_s=|u_s|^{2^\star_s-2}u_s, \quad u_s\in D^s_0(Ω),\quad 2^\star_s:=\frac{2N}{N-2s}, \end{equation*} where $s$ is any positive number, $Ω$ is either $\mathbb{R}^N$ or a smooth symmetric bounded domain, and $D^s_0(Ω)$ is the homogeneous Sobolev space. Depending on the kind of symmetry considered, solutions can be sign changing. We show that, up to a subsequence, a l.e.s.s. $u_s$ converges to a l.e.s.s. $u_{t}$ as $s$ goes to any $t>0$. In bounded domai…
Master equations with indefinite nonlinearities
本文研究了具有不定非线性性的分数热方程，证明了无正有界解，并提出新方法用于其他分数问题研究。
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