On existence and nonexistence of positive solutions of an elliptic system with coupled terms
Yayun Li Yutian Lei
Communications on Pure & Applied Analysis 2018, 17(5): 1749-1764 doi: 10.3934/cpaa.2018083
This paper is concerned with the elliptic system
$\begin{cases}-{\triangle u} = (q+1)u^qv^{p+1},~~ u>0~ in~ R^n,\\-{\triangle v} = (p+1)v^pu^{q+1},~~ v>0~in ~R^n,\end{cases}$
$ n ≥ 3 $
$ p,q>0 $
$ \max\{p,q\} ≥ 1 $
. We discuss the nonexistence of positive solutions in subcritical case and stable solutions in supercritical case, the necessary and sufficient conditions of classification in the critical case, and the Joseph-Lundgren-type condition for existence of local stable solutions.
keywords: Elliptic system Liouville theorem classification stable solutions Joseph-Lundgren condition
On the integral systems with negative exponents
Yutian Lei
Discrete & Continuous Dynamical Systems - A 2015, 35(3): 1039-1057 doi: 10.3934/dcds.2015.35.1039
This paper is concerned with the integral system $$\left \{ \begin{array}{ll} &u(x)=\int_{R^n}\frac{|x-y|^\lambda dy}{v^q(y)},\quad u>0~in~R^n,\\ &v(x)=\int_{R^n}\frac{|x-y|^\lambda dy}{u^p(y)},\quad v>0~in~R^n, \end{array} \right. $$ where $n \geq 1$, $p,q,\lambda \neq 0$. Such an integral system appears in the study of the conformal geometry. We obtain several necessary conditions for the existence of the $C^1$ positive entire solutions, particularly including the critical condition $$ \frac{1}{p-1}+\frac{1}{q-1}=\frac{\lambda}{n}, $$ which is the necessary and sufficient condition for the invariant of the system and some energy functionals under the scaling transformation. The necessary condition $\frac{1}{p-1}+\frac{1}{q-1}=\frac{\lambda}{n}$ can be relaxed to another weaker one $\min\{p,q\}>\frac{n+\lambda}{\lambda}$ for the system with double bounded coefficients. In addition, we classify the radial solutions in the case of $p=q$ as the form $$ u(x)=v(x)=a(b^2+|x-x_0|^2)^{\frac{\lambda}{2}} $$ with $a,b>0$ and $x_0 \in R^n$. Finally, we also deduce some analogous necessary conditions of existence for the weighted system.
keywords: classification of radial solutions. asymptotic behavior conformal invariant Singular integral equation
Wolff type potential estimates and application to nonlinear equations with negative exponents
Yutian Lei
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 2067-2078 doi: 10.3934/dcds.2015.35.2067
In this paper, we are concerned with the positive continuous entire solutions of the Wolff type integral equation $$ u(x)=c(x)W_{\beta,\gamma}(u^{-p})(x), \quad u>0 ~in~ R^n, $$ where $n \geq 1$, $p>0$, $\gamma>1$, $\beta>0$ and $\beta\gamma \neq n$. In addition, $c(x)$ is a double bounded function. Such an integral equation is related to the study of the conformal geometry and nonlinear PDEs, such as $\gamma$-Laplace equations and $k$-Hessian equations with negative exponents. By some Wolff type potential integral estimates, we obtain the asymptotic rates and the integrability of positive solutions, and discuss the existence and nonexistence results of the radial solutions.
keywords: asymptotic behavior. Wolff potential $\gamma$-Laplace equation $k$-Hessian equation conformal geometry
Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality
Yutian Lei Zhongxue Lü
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 1987-2005 doi: 10.3934/dcds.2013.33.1987
This paper is concerned with the symmetry results for the $2k$-order singular Lane-Emden type partial differential system $$ \left\{\begin{array}{ll} (-\Delta)^k(|x|^{\alpha}u(x)) =|x|^{-\beta} v^{q}(x), \\ (-\Delta)^k(|x|^{\beta}v(x)) =|x|^{-\alpha} u^p(x), \end{array} \right. $$ and the weighted Hardy-Littlewood-Sobolev type integral system $$ \left \{ \begin{array}{l} u(x) = \frac{1}{|x|^{\alpha}}\int_{R^{n}} \frac{v^q(y)}{|y|^{\beta}|x-y|^{\lambda}} dy\\ v(x) = \frac{1}{|x|^{\beta}}\int_{R^{n}} \frac{u^p(y)}{|y|^{\alpha}|x-y|^{\lambda}} dy. \end{array} \right. $$ Here $x \in R^n \setminus \{0\}$. We first establish the equivalence of this integral system and an fractional order partial differential system, which includes the $2k$-order PDE system above. For the integral system, we prove that the positive locally bounded solutions are symmetric and decreasing about some axis by means of the method of moving planes in integral forms introduced by Chen-Li-Ou. In addition, we also show that the integrable solutions are locally bounded. Thus, the equivalence implies the positive solutions of the PDE system, particularly including the higher integer-order PDE system, also have the corresponding properties.
keywords: weighted Hardy-Littlewood-Sobolev inequality method of moving planes axisymmetry. Higher-order Lane-Emden system
Positive solutions of integral systems involving Bessel potentials
Yutian Lei
Communications on Pure & Applied Analysis 2013, 12(6): 2721-2737 doi: 10.3934/cpaa.2013.12.2721
This paper is concerned with integral systems involving the Bessel potentials. Such integral systems are helpful to understand the corresponding PDE systems, such as some static Shrödinger systems with the critical and the supercritical exponents. We use the lifting lemma on regularity to obtain an integrability interval of solutions. Since the Bessel kernel does not have singularity at infinity, we extend the integrability interval to the whole $[1,\infty]$. Next, we use the method of moving planes to prove the radial symmetry for the positive solution of the system. Based on these results, by an iteration we obtain the estimate of the exponential decay of those solutions near infinity. Finally, we discuss the uniqueness of the positive solution of PDE system under some assumption.
keywords: method of moving planes integrability intervals Integral equations Bessel potential decay rate. Hardy-Littlewood-Sobolev inequality radial symmetry
Asymptotic behavior for solutions of some integral equations
Yutian Lei Chao Ma
Communications on Pure & Applied Analysis 2011, 10(1): 193-207 doi: 10.3934/cpaa.2011.10.193
In this paper we study the asymptotic behavior of the positive solutions of the following system of Euler-Lagrange equations of the Hardy-Littlewood-Sobolev type in $R^n$

$u(x) = \frac{1}{|x|^{\alpha}}\int_{R^n} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}} dy $,

$ v(x) = \frac{1}{|x|^{\beta}}\int_{R^n} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}}dy. $

We obtain the growth rate of the solutions around the origin and the decay rate near infinity. Some new cases beyond the work of C. Li and J. Lim [17] are studied here. In particular, we remove some technical restrictions of [17], and thus complete the study of the asymptotic behavior of the solutions for non-negative $\alpha$ and $\beta$.

keywords: Integral equations weighted Hardy-Littlewood-Sobolev inequality. singularities asymptotic analysis
Liouville theorems and classification results for a nonlocal Schrödinger equation
Yutian Lei
Discrete & Continuous Dynamical Systems - A 2018, 38(11): 5351-5377 doi: 10.3934/dcds.2018236
In this paper, we study the existence and the nonexistence of positive classical solutions of the static Hartree-Poisson equation
$-Δ u = pu^{p-1}(|x|^{2-n}*u^p),\;\; u>0 \;\;in\;\; R^n, $
$n ≥ 3$
$p≥ 1$
. The exponents of the Serrin type, the Sobolev type and the Joseph-Lundgren type play the critical roles as in the study of the Lane-Emden equation. First, we prove that the equation has no positive solution when
$1 ≤ p <\frac{n+2}{n-2}$
by means of the method of moving planes to the following system
$\left\{ \begin{array}{l} - \Delta u = \sqrt p {u^{p - 1}}v,\;\;u > 0\;\;in\;\;{R^n},\\ - \Delta v = \sqrt p {u^p},\;\;v > 0\;\;in\;\;{R^n}.\end{array} \right.$
$p = \frac{n+2}{n-2}$
, all the positive solutions can be classified as
$u(x) = c(\frac{t}{t^2+|x-x^*|^2})^{\frac{n-2}{2}}$
with the help of an integral system involving the Newton potential, where
$c, t$
are positive constants, and
$x^* ∈ R^n$
. In addition, we also give other equivalent conditions to classify those positive solutions. When
, by the shooting method and the Pohozaev identity, we find radial solutions for the system. In particular, the equation has a radial solution decaying with slow rate
. Finally, we point out that the equation has positive stable solutions if and only if
$p ≥ 1+\frac{4}{n-4-2\sqrt{n-1}}$
keywords: Hartree-Poisson equation Liouville theorem classification critical exponent stable solution
On finite energy solutions of fractional order equations of the Choquard type
Yutian Lei
Discrete & Continuous Dynamical Systems - A 2019, 39(3): 1497-1515 doi: 10.3934/dcds.2019064

Finite energy solutions are the important class of solutions of the Choquard equation. This paper is concerned with the regularity of weak finite energy solutions. For nonlocal fractional-order equations, an integral system involving the Riesz potential and the Bessel potential plays a key role. Applying the regularity lifting lemma to this integral system, we can see that some weak integrable solution has the better regularity properties. In addition, we also show the relation between such an integrable solution and the finite energy solution. Based on these results, we prove that the weak finite energy solution is also the classical solution under some conditions. Finally, we point out that the least energy with the critical exponent can be represented by the sharp constant of some inequality of Sobolev type though the ground state solution cannot be found.

keywords: Choquard equation finite energy solution integrable solution Bessel potential
Decay estimation for positive solutions of a $\gamma$-Laplace equation
Yutian Lei Congming Li Chao Ma
Discrete & Continuous Dynamical Systems - A 2011, 30(2): 547-558 doi: 10.3934/dcds.2011.30.547
In this paper, we study the properties of the positive solutions of a $\gamma$-Laplace equation in $R^n$

-div$(|\nabla u|^{\gamma-2}\nabla u) =K u^p$,

Here $1<\gamma<2$, $n>\gamma$, $p=\frac{(\gamma-1)(n+\gamma)}{n-\gamma}$ and $K(x)$ is a smooth function bounded by two positive constants. First, the positive solution $u$ of the $\gamma$-Laplace equation above satisfies an integral equation involving a Wolff potential. Based on this, we estimate the decay rate of the positive solutions of the $\gamma$-Laplace equation at infinity. A new method is introduced to fully explore the integrability result established recently by Ma, Chen and Li on Wolff type integral equations to derive the decay estimate.

keywords: asymptotic analysis. decay rate Wolff potential $\gamma$-Laplace equation integral equation

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