189 research outputs found
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
. 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 , , and with . In the special case when
and , 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 -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 's. Recently, a
degree theory approach to shooting method for this broad class of problems is
introduced in \cite{LiarXiv13} for nonnegative '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 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 and
show that 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 <
- β¦