arXiv subjects
Publications and source records attributed to Congming Li.
In this short article, we state a Hopf type lemma for fractional equations and the outline of its proof. We believe that it will become a powerful tool in applying the method of moving planes on fractional equations to obtain qualitative properties of solutions.
In this paper, we consider nonlinear equations involving the fractional p-Laplacian $$ (-\lap)_p^s u(x)) \equiv C_{n,s,p} PV \int_{\mathbb{R}^n} \frac{|u(x)-u(y)|^{p-2}[u(x)-u(y)]}{|x-z|^{n+ps}} dz= f(x,u).$$ We prove a {\em maximum principle for anti-symmetric functions} and obtain other key ingredients for carrying on the method of moving planes, such as {\em a key boundary estimate lemma}. Then we establish radial symmetry and monotonicity for positive solutions to semilinear equations involving the fractional p-Laplacian in a unit ball and in the whole space. We believe that the methods developed here can be applied to a variety of problems involving nonlinear nonlocal operators.
In this paper, we develop a direct method of moving planes for the fractional Laplacian. Instead of conventional extension method introduced by Caffarelli and Silvestre, we work directly on the non-local operator. Using the integral defining the fractional Laplacian, by an elementary approach, we first obtain the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space, such as strong maximum principles for anti-symmetric functions, narrow region principles, and decay at infinity. Then, using a simple example, a semi-linear equation involving the fractional Laplacian, we illustrate how this new method of moving planes can be employed to obtain symmetry and non-existence of positive solutions. We firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities.
In this paper, we consider equations involving fully nonlinear nonlocal operators $$F_α(u(x)) \equiv C_{n,α} PV \int_{\mathbb{R}^n} \frac{G(u(x)-u(z))}{|x-z|^{n+α}} dz= f(x,u).$$ We prove a maximum principle and obtain key ingredients for carrying on the method of moving planes, such as narrow region principle and decay at infinity. Then we establish radial symmetry and monotonicity for positive solutions to Dirichlet problems associated to such fully nonlinear fractional order equations in a unit ball and in the whole space, as well as non-existence of solutions on a half space. We believe that the methods develop here can be applied to a variety of problems involving fully nonlinear nonlocal operators. We also investigate the limit of this operator as $α\rightarrow 2$ and show that $$F_α(u(x)) \rightarrow a(-Δu(x)) + b |\bigtriangledown u(x)|^2 .$$
In this paper, we present a necessary and sufficient condition to the Lane-Emden conjecture. This condition is an energy type of integral estimate on solutions to subcritical Lane-Emden system. To approach the long standing and interesting conjecture, we believe that one plausible path is to refocus on establishing this energy type estimate.
In this paper, we study some qualitative properties of Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized into three cases: critical, supercritical and subcritical. The critical case, the well known original HLS system, corresponds to the Euler-Lagrange equations of the fundamental HLS inequality. In each case, we give a brief survey on some important results and useful methods. Some simplifications and extensions based on somewhat more direct and intuitive ideas are presented. Also, a few new qualitative properties are obtained and several open problems are raised for future research.
In this paper, we develop a direct {\em blowing-up and rescaling} argument for a nonlinear equation involving the fractional Laplacian operator. Instead of using the conventional extension method introduced by Caffarelli and Silvestre, we work directly on the nonlocal operator. Using the integral defining the nonlocal elliptic operator, by an elementary approach, we carry on a {\em blowing-up and rescaling} argument directly on nonlocal equations and thus obtain a priori estimates on the positive solutions for a semi-linear equation involving the fractional Laplacian. We believe that the ideas introduced here can be applied to problems involving more general nonlocal operators.
In this paper, we consider $α$-harmonic functions in the half space $\mathbb{R}^n_+$: \begin{equation} \left\{\begin{array}{ll} (-Δ)^{α/2} u(x)=0,~u(x)>0, & x\in\mathbb{R}^n_+, \\ u(x)\equiv 0, & x\notin \mathbb{R}^{n}_{+}. \end{array}\right. \end{equation} We prove that all the solutions have to assume the form \begin{equation} u(x)=\left\{\begin{array}{ll}Cx_n^{α/2}, & \qquad x\in\mathbb{R}^n_+, \\ 0, & \qquad x\notin\mathbb{R}^{n}_{+}, \end{array}\right. \label{2} \end{equation} for some positive constant $C$.
In this paper, we apply the moving plane method to the following high order degenerate elliptic equation,\begin{equation*} (-A)^p u=u^α\text{ in } \mathbb R^{n+1}_+,n\geq 1, \end{equation*}where the operator $A=y\partial_y^2+a\partial_y+Δ_x,a\geq 1$. We get a Liouville theorem for subcritical case and classify the solutions for the critical case.
In this paper, we study the existence of solution to a nonlinear system: \begin{align} \left\{\begin{array}{cl} -Δu_{i} = f_{i}(u) & \text{in } \mathbb{R}^n, u_{i} > 0 & \text{in } \mathbb{R}^n, \, i = 1, 2,\cdots, L % u_{i}(x) \rightarrow 0 & \text{uniformly as } |x| \rightarrow \infty \end{array} \right. \end{align} for sign changing nonlinearities $f_i$'s. Recently, a degree theory approach to shooting method for this broad class of problems is introduced in \cite{LiarXiv13} for nonnegative $f_i$'s. However, many systems of nonlinear Schödinger type involve interaction with undetermined sign. Here, based on some new dynamic estimates, we are able to extend the degree theory approach to systems with sign-changing source terms.
In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, \begin{equation} \sum_{i,j,i\neq j}\frac{f_{i}g_{j}}{\mid i-j\mid^{n-α}}\leq C_{r,s,α} |f|_{l^r} |g|_{l^s}, \end{equation}where $i,j\in \mathbb Z^n$, $r,s>1$, $0<α 2$.
Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case: μ=n. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: μ=n and p=q, by limiting the inequality on a finite domain. The best constant in the inequality and its corresponding solution, the optimizer, are studied. First, we obtain a sharp estimate for the best constant. Then for the optimizer, we prove the uniqueness and a symmetry property. This is achieved by proving that the corresponding Euler-Lagrange equation has a unique nontrivial nonnegative critical point. Also, by using a discrete version of maximum principle, we prove certain monotonicity of this optimizer.
In this paper, we study the asymptotic behavior of positive solutions of the nonlinear differential systems of Lane-Emden type $2k$-order equations $$\{{array}{l} (-Δ)^k u=v^q,u>0 \quad in ~R^n, (-Δ)^k v=u^p,v>0 \quad in ~R^n, {array}. $$ and the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations $$ \{{array}{l} u(x)=\displaystyle\int_{R^n}\frac{v^q(y)dy}{|x-y|^{n-α}},u>0 \quad in ~R^n, v(x)=\displaystyle\int_{R^n}\frac{u^p(y)dy}{|x-y|^{n-α}},u>0 \quad in ~R^n. {array}. $$ Such an integral system is related to the study the extremal functions of the HLS inequality. We point out that the bounded solutions $u,v$ converge to zero either with the fast decay rates or with the slow decay rates when $|x| \to \infty$ under some assumptions. In addition, we also find a criterion to distinguish the fast and the slow decay rates: if $u,v$ are the integrable solutions (i.e. $(u,v) \in L^{r_0}(R^n) \times L^{s_0}(R^n)$), then they decay fast; if the bounded solutions $u,v$ are not the integrable solutions (i.e. $(u,v) \not\in L^{r_0}(R^n) \times L^{s_0}(R^n)$), then they decay almost slowly. Here, for the HLS type system, $r_0=\frac{n(pq-1)}{α(q+1)}$, $s_0=\frac{n(pq-1)}{α(p+1)}$; and for the Lane-Emden type system, $r_0,s_0$ are still the forms above where $α$ is replaced by $2k$.
The classical shooting-method is about finding a suitable initial shooting positions to shoot to the desired target. The new approach formulated here, with the introduction and the analysis of the `target map' as its core, naturally connects the classical shooting-method to the simple and beautiful topological degree theory. We apply the new approach, to a motivating example, to derive the existence of global positive solutions of the Hardy-Littlewood-Sobolev (also known as Lane-Emden) type system: [{{aligned} &(-\triangle)^ku(x) = v^p(x), \,\, u(x)>0 \quad\text{in}\quad\mathbb{R}^n, & (-\triangle)^k v(x) =u^q(x), \,\, v(x)>0 \quad\text{in}\quad\mathbb{R}^n, p, q>0, {aligned}.] in the critical and supercritical cases $\frac{1}{p+1}+\frac{1}{q+1}\leq\frac{n-2k}{n}$. Here we derive the existence with the computation of the topological degree of a suitably defined target map. This and some other results presented in this article completely solved several long-standing open problems about the existence or non-existence of positive entire solutions.
In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type equations. These nonexistence results, known as Liouville type theorems, are fundamental in PDE theory and applications. A special iteration scheme, a new shooting method and some Pohozaev type identities in integral form as well as in differential form are created. Combining these new techniques with some observations and some critical asymptotic analysis, we establish the sharp criteria of Liouville type for our systems of nonlinear equations. Similar results are also derived for the system of Wolff type integral equations and the system of $γ$-Laplace equations. A dichotomy description in terms of existence and nonexistence for solutions with finite energy is also obtained.
In this paper, we study global well-posedness of the two-dimensional Keller-Segel model in Lebesgue space and Sobolev space. Recall that in the paper "Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space, J. Differential Equations, {252}(2012), 1213--1228", Kozono, Sugiyama & Wachi studied global well-posedness of $n$($\ge3$) dimensional Keller-Segel system and posted a question about the even local in time existence for the Keller-Segel system with $L^1(R^2)\times{L}^\infty(R^2)$ initial data. Here we give an affirmative answer to this question: in fact, we show the global in time existence and uniqueness for $L^1(R^2)\times{L}^{\infty}(R^2)$ initial data. Furthermore, we prove that for any $H^1_b(R^2) \times {H}^1(R^2)$ initial data with $H^1_b(R^2):=H^1(R^2)\cap{L}^\infty(R^2)$, there also exists a unique global mild solution to the parabolic-parabolic Keller-Segel model. The estimates of ${\sup_{t>0}}t^{1-\frac{n}{p}}\|u\|_{L^p}$ for $(n,p)=(2,\infty)$ and the introduced special half norm, i.e. $\sup_{t>0}t^{1/2}(1+t)^{-1/2}\|\nabla{v}\|_{L^\infty}$, are crucial in our proof.
In this paper, we prove that all the positive solutions for the PDE system (-Δ)^{k}u_{i} = f_{i}(u_{1},..., u_{m}), x \in R^{n}, i = 1, 2,..., m are super polyharmonic, i.e. (-Δ)^{j}u_{i} > 0, j = 1, 2,..., k - 1; i = 1, 2,...,m. To prove this important super polyharmonic property, we introduced a few new ideas and derived some new estimates. As an interesting application, we establish the equivalence between the integral system u_{i}(x) = \int_{R^{n}} \frac{1}{|x - y|^{n-α}}f_{i}(u_{1}(y),..., u_{m}(y))dy, x \in R^{n} and PDE system when α? = 2k < n
We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei in [13] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier-Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove the global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.