Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLiouville theorem and isolated singularity of fractional Laplacian system with critical exponents
59
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This paper is devoted to the fractional Laplacian system with critical exponents. We use the method of moving sphere to derive a Liouville Theorem, and then prove the solutions in R^n{0} are radially symmetric and monotonically decreasing radially. Together with blow up analysis and the Pohozaev integral, we get the upper and lower bound of the local solutions in B_1{0}. Our results is an extension of the classical work by Caffarelli et al [6, 7], Chen et al[16]
rate research
Read More
In this paper, we establish a new improved Sobolev inequality based on a weighted Morrey space. To be precise, there exists $C=C(n,m,s,alpha)>0$ such that for any $u,v in {dot{H}}^s(mathbb{R}^{n})$ and for any $theta in (bar{theta},1)$, it holds that begin{equation} label{eq0.3} Big( int_{ mathbb{R}^{n} } frac{ |(uv)(y)|^{frac{2^*_{s}(alpha)}{2} } } { |y|^{alpha} } dy Big)^{ frac{1}{ 2^*_{s} (alpha) }} leq C ||u||_{{dot{H}}^s(mathbb{R}^{n})}^{frac{theta}{2}} ||v||_{{dot{H}}^s(mathbb{R}^{n})}^{frac{theta}{2}} ||(uv)||^{frac{1-theta}{2}}_{ L^{1,n-2s+r}(mathbb{R}^{n},|y|^{-r}) }, end{equation} where $s !in! (0,1)$, $0!<!alpha!<!2s!<!n$, $2s!<!m!<!n$, $bar{theta}=max { frac{2}{2^*_{s}(alpha)}, 1-frac{alpha}{s}cdotfrac{1}{2^*_{s}(alpha)}, frac{2^*_{s}(alpha)-frac{alpha}{s}}{2^*_{s}(alpha)-frac{2alpha}{m}} }$, $r=frac{2alpha}{ 2^*_{s}(alpha) }$ and $y!=!(y,y) in mathbb{R}^{m} times mathbb{R}^{n-m}$. By using mountain pass lemma and (ref{eq0.3}), we obtain a nontrivial weak solution to a doubly critical system involving fractional Laplacian in $mathbb{R}^{n}$ with partial weight in a direct way. Furthermore, we extend inequality (ref{eq0.3}) to more general forms on purpose of studying some general systems with partial weight, involving p-Laplacian especially.
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 $alpha$-stable process and probabilistic representation of the super-solutions.
We study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $zetainpartialOmegacup{infty}$ of the quasilinear elliptic equations $$-text{div}(| abla u|_A^{p-2}A abla u)+V|u|^{p-2}u =0quadtext{in } Omegasetminus{zeta},$$ where $Omega$ is a domain in $mathbb{R}^d$ ($dgeq 2$), and $A=(a_{ij})in L_{rm loc}^{infty}(Omega;mathbb{R}^{dtimes d})$ is a symmetric and locally uniformly positive definite matrix. The potential $V$ lies in a certain local Morrey space (depending on $p$) and has a Fuchsian-type isolated singularity at $zeta$.
In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation begin{equation*} (-Delta)^{frac{alpha}{2}} u=lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{alpha}-2}u, quadtext{in},,Omega, u=0,text{on},,partialOmega, end{equation*} where $Omegasubset mathbb{R}^{N}(Ngeq 2)$ is a bounded domain with smooth boundary, $0<alpha<2$, $(-Delta)^{frac{alpha}{2}}$ stands for the fractional Laplacian operator, $fin C(Omegatimesmathbb{R},mathbb{R})$ may be sign changing and $lambda$ is a positive parameter. We will prove that there exists $lambda_{*}>0$ such that the problem has at least two positive solutions for each $lambdain (0,,,lambda_{*})$. In addition, the concentration behavior of the solutions are investigated.
In this paper, we prove several Liouville type results for a nonlinear equation involving infinity Laplacian with gradient of the form $$Delta^gamma_infty u + q(x)cdot abla{u} | abla{u}|^{2-gamma} + f(x, u),=,0quad text{in}; mathbb{R}^d,$$ where $gammain [0, 2]$ and $Delta^gamma_infty$ is a $(3-gamma)$-homogeneous operator associated with the infinity Laplacian. Under the assumptions $liminf_{|x|toinfty}lim_{sto0}f(x,s)/s^{3-gamma}>0$ and $q$ is a continuous function vanishing at infinity, we construct a positive bounded solution to the equation and if $f(x,s)/s^{3-gamma}$ decreasing in $s$, we also obtain the uniqueness. While, if $limsup_{|x|toinfty}sup_{[delta_1,delta_2]}f(x,s)<0$, then nonexistence result holds provided additionally some suitable conditions. To this aim, we develop new technique to overcome the degeneracy of infinity Laplacian and nonlinearity of gradient term. Our approach is based on a new regularity result, the strong maximum principle, and Hopfs lemma for infinity Laplacian involving gradient and potential. We also construct some examples to illustrate our results. We further study the related Dirichlet principal eigenvalue of the corresponding nonlinear operator $$Delta^gamma_infty u + q(x)cdot abla{u} | abla{u}|^{2-gamma} + c(x)u^{3-gamma},$$ in smooth bounded domains, which may be considered as of independent interest. Our results could be seen as the extension of Liouville type results obtained by Savin [48] and Ara{u}jo et. al. [1] and a counterpart of the uniqueness obtained by Lu and Wang [39,40] for sign-changing $f$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
