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

A direct method of moving planes for the fractional Laplacian

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.Comment: 36 page

Super Polyharmonic Property of Solutions for PDE Systems and Its Applications

In this paper, we prove that all the positive solutions for the PDE system (-\Delta)^{k}u_{i} = f_{i}(u_{1},..., u_{m}), x \in R^{n}, i = 1, 2,..., m are super polyharmonic, i.e. (-\Delta)^{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-\alpha}}f_{i}(u_{1}(y),..., u_{m}(y))dy, x \in R^{n} and PDE system when \alpha? = 2k <