A degree theory approach for the shooting method

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

Uniqueness of positive bound states to Schrodinger systems with critical exponents

We prove the uniqueness for the positive solutions of the following elliptic systems: \begin{eqnarray*} \left\{\begin{array}{ll} - \lap (u(x)) = u(x)^{\alpha}v(x)^{\beta} - \lap (v(x)) = u(x)^{\beta} v(x)^{\alpha} \end{array} \right. \end{eqnarray*} Here $x\in R^n$, $n\geq 3$, and $1\leq \alpha, \beta\leq \frac{n+2}{n-2}$ with $\alpha+\beta=\frac{n+2}{n-2}$. In the special case when $n=3$ and $\alpha =2, \beta=3$, the systems come from the stationary Schrodinger system with critical exponents for Bose-Einstein condensate. As a key step, we prove the radial symmetry of the positive solutions to the elliptic system above with critical exponents

Sharp criteria of Liouville type for some nonlinear systems

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 $\gamma$-Laplace equations. A dichotomy description in terms of existence and nonexistence for solutions with finite energy is also obtained

Shooting Method with Sign-Changing Nonlinearity

In this paper, we study the existence of solution to a nonlinear system: \begin{align} \left\{\begin{array}{cl} -\Delta u_{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 Sch\"odinger 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

A Hopf type lemma for fractional equations

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.Comment: 7 page

An Extended Discrete Hardy-Littlewood-Sobolev Inequality

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case: \mu=n. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: \mu=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

Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions

In this paper, we consider equations involving fully nonlinear nonlocal operators $$F_{\alpha}(u(x)) \equiv C_{n,\alpha} PV \int_{\mathbb{R}^n} \frac{G(u(x)-u(z))}{|x-z|^{n+\alpha}} dz= f(x,u).$$ 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 $\alpha \rightarrow 2$ and show that $$F_{\alpha}(u(x)) \rightarrow a(-\Delta u(x)) + b |\bigtriangledown u(x)|^2 .$$Comment: 27 pages. arXiv admin note: text overlap with arXiv:1411.169

A direct blowing-up and rescaling argument on the fractional Laplacian equation

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

Global Well-posedness of the Parabolic-parabolic Keller-Segel Model in L1(R2)×L∞(R2)L^{1}(R^2)\times{L}^{\infty}(R^2) and Hb1(R2)×H1(R2)H^1_b(R^2)\times{H}^1(R^2)

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

Decay properties of the Hardy-Littlewood-Sobolev systems of the Lane-Emden type

In this paper, we study the asymptotic behavior of positive solutions of the nonlinear differential systems of Lane-Emden type $2k$-order equations $$\{{array}{l} (-\Delta)^k u=v^q,u>0 \quad in ~R^n, (-\Delta)^k v=u^p,v>0 \quad in ~R^n, {array}.$$ and the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations $$\{{array}{l} u(x)=\displaystyle\int_{R^n}\frac{v^q(y)dy}{|x-y|^{n-\alpha}},u>0 \quad in ~R^n, v(x)=\displaystyle\int_{R^n}\frac{u^p(y)dy}{|x-y|^{n-\alpha}},u>0 \quad in ~R^n. {array}.$$ 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, $r_0=\frac{n(pq-1)}{\alpha(q+1)}$, $s_0=\frac{n(pq-1)}{\alpha(p+1)}$; and for the Lane-Emden type system, $r_0,s_0$ are still the forms above where $\alpha$ is replaced by $2k$.Comment: 24 page