Radial Symmetry Of Positive Solutions Research Articles

Radial symmetry of positive solutions for a tempered fractional p-Laplacian system

Radial symmetry of positive solutions for a tempered fractional p-Laplacian system

The radial symmetry of positive solutions for semilinear problems involving weighted fractional Laplacians

The radial symmetry of positive solutions for semilinear problems involving weighted fractional Laplacians

On Semilinear Elliptic Equations with Hardy-Leray Potentials

This paper is concerned with a semilinear elliptic equation with the Hardy-Leray potential. We employ the method of moving planes to prove the radial symmetry of positive solutions. Based on this result, we obtain the Liouville theorem in subcritical case. In addition, we find special radial solutions in critical case. All the properties above are similar to the corresponding results of the Lane-Emden equation.

Radial symmetry for a generalized nonlinear fractional p-Laplacian problem

This paper first introduces a generalized fractional p-Laplacian operator (–Δ)sF;p. By using the direct method of moving planes, with the help of two lemmas, namely decay at infinity and narrow region principle involving the generalized fractional p-Laplacian, we study the monotonicity and radial symmetry of positive solutions of a generalized fractional p-Laplacian equation with negative power. In addition, a similar conclusion is also given for a generalized Hénon-type nonlinear fractional p-Laplacian equation.

Existence and symmetry of solutions to 2-D Schrödinger–Newton equations

In this paper, we consider the following 2-D Schr\"{o}dinger-Newton equations \begin{eqnarray*} -\Delta u+a(x)u+\frac{\gamma}{2\pi}\left(\log(|\cdot|)*|u|^p\right){|u|}^{p-2}u=b{|u|}^{q-2}u \qquad \text{in} \,\,\, \mathbb{R}^{2}, \end{eqnarray*} where $a\in C(\mathbb{R}^{2})$ is a $\mathbb{Z}^{2}$-periodic function with $\inf_{\mathbb{R}^{2}}a>0$, $\gamma>0$, $b\geq0$, $p\geq2$ and $q\geq 2$. By using ideas from \cite{CW,DW,Stubbe}, under mild assumptions, we obtain existence of ground state solutions and mountain pass solutions to the above equations for $p\geq2$ and $q\geq2p-2$ via variational methods. The auxiliary functional $J_{1}$ plays a key role in the cases $p\geq3$. We also prove the radial symmetry of positive solutions (up to translations) for $p\geq2$ and $q\geq 2$. The corresponding results for planar Schr\"{o}dinger-Poisson systems will also be obtained. Our theorems extend the results in \cite{CW,DW} from $p=2$ and $b=1$ to general $p\geq2$ and $b\geq0$.

A unified approach to symmetry for semilinear equations associated to the Laplacian in [formula omitted

We show radial symmetry of positive solutions to the Hénon equation −Δu=|x|−ℓuq in RN∖{0}, where ℓ≥0, q>0 and satisfy further technical conditions. A new ingredient is a maximum principle for open subsets of a half space. It allows to apply the Moving Plane Method once a slow decay of the solution at infinity has been established, that is lim|x|→∞⁡|x|γu(x)=L, for some numbers γ∈(0,N−2) and L>0. Moreover, some examples of non-radial solutions are given for q>N+1N−3 and N≥4. We also establish radial symmetry for related and more general problems in RN and RN∖{0}.

Symmetry of positive solutions for Choquard equations with fractional [formula omitted]-Laplacian

This paper deals with the following Choquard equations with fractional p-Laplacian −Δpsu=Cn,t|x|2t−n∗uquq−1inRn,u>0onRn,where 0<s,t<1, 2<p<∞, p−1<q<∞ and n≥2. By constructing a decay at infinity theorem and a narrow region principle, we establish the radial symmetry of positive solutions based on the generalized direct method of moving planes.

Radial symmetry for positive solutions of fractional p-Laplacian equations via constrained minimization method

The aim of this paper is to investigate a class of fractional p-Laplacian equations. We obtain existence and symmetry results for solutions in the fractional Sobolev space Ws,p(Rn) by rearrangement of its corresponding constrained minimization. Our results are in accordance with those for the classical p-Laplacian equations and fractional Schrödinger equations.

Regularity and radial symmetry of positive solutions for a higher order elliptic system

We discuss the properties of solutions for the following elliptic partial differential equations system in R n , $$\begin{cases}(-\triangle)^{\frac{\alpha}{2}} u&= u^{p_{1}} v^{p_{2}}\\(-\triangle)^{\frac{\alpha}{2}} v &= u^{q_{1}} v^{q_{2}}\end{cases}$$ where 0 < α < n, p i and q i (i = 1, 2) satisfy some suitable assumptions. Due to equivalence between the PDEs system and a given integral system, we prove the radial symmetry and regularity of positive solutions to the PDEs system via the method of moving plane in integral forms and Regularity Lifting Lemma. For the special case, when p 1 + p 2 = q 1 + q 2 = \(\frac{n+\alpha}{n-\alpha}\), we classify the solutions of the PDEs system.

Existence and uniqueness of positive solutions to three coupled nonlinear Schrödinger equations

In this paper, we consider existence and uniqueness of positive solutions to three coupled nonlinear Schrodinger equations which appear in nonlinear optics. We use the behaviors of minimizing sequences for a bound to obtain the existence of positive solutions for three coupled system. To prove the uniqueness of positive solutions, we use the radial symmetry of positive solutions to transform the system into an ordinary differential system, and then integrate the system. In particular, for N = 1, we prove the uniqueness of positive solutions when 0 ≤ β = μ1 = μ2 = μ3 or β > max{μ1, μ2, μ3}.

Symmetry of positive solutions of fractional Laplacian equation and system with Hardy–Sobolev exponent on the unit ball

In this work, we consider a class of Hardy–Sobolev differential equations and systems involving the fractional Laplacian on the unit ball. We first show the differential equations and systems are equivalent to some integral equations and systems, respectively. Then applying the method of moving planes in the integral forms, we prove the radial symmetry of positive solutions.

Radial symmetry of positive solutions for nonlinear elliptic boundary value problems

The aim of this paper is to study the symmetry properties of positive solutions of nonlinear elliptic boundary value problems of type$$\begin{gathered}\Delta u+f(|x|, u, \nabla u)=0 \text { in } R^n . \\u(x) \rightarrow 0 \text { as }|x| \rightarrow \infty\end{gathered}$$We employ the moving plane method based on maximum principle on unbounded domains to obtain the result on symmetry.

RADIAL SYMMETRY OF POSITIVE SOLUTIONS TO EQUATIONS INVOLVING THE FRACTIONAL LAPLACIAN

The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem [Formula: see text] where (-Δ)αdenotes the fractional Laplacian, α ∈ (0, 1), and B1denotes the open unit ball centered at the origin in ℝNwith N ≥ 2. The function f : [0, ∞) → ℝ is assumed to be locally Lipschitz continuous and g : B1→ ℝ is radially symmetric and decreasing in |x|. In the second place we consider radial symmetry of positive solutions for the equation [Formula: see text] with u decaying at infinity and f satisfying some extra hypothesis, but possibly being non-increasing.Our third goal is to consider radial symmetry of positive solutions for system of the form [Formula: see text] where α1, α2∈(0, 1), the functions f1and f2are locally Lipschitz continuous and increasing in [0, ∞), and the functions g1and g2are radially symmetric and decreasing. We prove our results through the method of moving planes, using the recently proved ABP estimates for the fractional Laplacian. We use a truncation technique to overcome the difficulty introduced by the non-local character of the differential operator in the application of the moving planes.

Asymptotic behavior of positive solutions of a nonlinear integral system

In this paper, we study the properties of positive solutions of an integral system in R n { u ( x ) = ∫ R n v p ( y ) u q ( y ) | x − y | n − α | y | − σ d y v ( x ) = ∫ R n v q ( y ) u p ( y ) | x − y | n − α | y | − σ d y . We prove the radial symmetry of positive solutions by means of the method of moving planes in integral forms which is established by Chen–Li–Ou. In addition, by using a regularity lifting lemma, we obtain an integrability result. Under the assumption α + ( p + q + 1 ) σ ≥ 0 , we prove that those solutions are the ground states. Finally, we estimate the decay rates of such solutions when | x | → ∞ .

Radial symmetry of positive solutions for semilinear elliptic equations in the unit ball via elliptic and hyperbolic geometry

Let n ∈ N with n ⩾ 2 , a ∈ ( − 1 , 0 ) ∪ ( 0 , 1 ] and f : ( 0 , 1 ) × ( 0 , ∞ ) → R such that for each u ∈ ( 0 , ∞ ) , r ↦ ( 1 + a r 2 ) ( n + 2 ) / 2 f ( r , ( 1 + a r 2 ) − ( n − 2 ) / 2 u ) : ( 0 , 1 ) → R is nonincreasing. We show that each positive solution of Δ u + f ( | x | , u ) = 0 in B , u = 0 on ∂ B is radially symmetric, where B is the open unit ball in R N .

Classification of Positive Solitary Solutions of the Nonlinear Choquard Equation

In this paper, we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary Choquard equation $$\Delta u-u+2u\left(\frac{1}{|x|}*|u|^2\right)=0, \quad u\in H^1(\mathbb{R}^3),$$ which can be considered as a certain approximation of the Hartree–Fock theory for a one component plasma as explained in Lieb and Lieb–Simon’s papers starting from 1970s. We first prove that all the positive solutions of this equation must be radially symmetric and monotone decreasing about some fixed point. Interestingly, to use the new method of moving planes introduced by Chen–Li–Ou, we deduce the problem into an elliptic system. As a key step, we transform this differential system into a system of integral equations with the help of Riesz and Bessel potentials, and then use the method of a moving plane in an integral form. Next, using radial symmetry, we deduce the uniqueness result from Lieb’s work. Our argument can be adapted well to study the radial symmetry of positive solutions of the equation in the generalized form $$u=K_1*F\left(u,K_2*u\right)$$ .

Nonexistence and Radial Symmetry of Positive Solutions of Semilinear Elliptic Systems

Nonexistence and radial symmetry of positive solutions for a class of semilinear elliptic systems are considered via the method of moving spheres.

An integral system and the Lane-Emden conjecture

We consider the system of integral equations in $R^n$: <p align="center"> $ u(x) = \int_{R^{n}} \frac{1}{|x - y|^{n-\mu}} v^q (y) dy$<BR> $ v(x) = \int_{R^{n}} \frac{1}{|x - y|^{n-\mu}} u^p(y) dy$<br> <p align="left" class="times"> with $0 < \mu < n$. Under some integrability conditions, we obtain radial symmetry of positive solutions by using <i>the method of moving planes in integral forms</i>. In the special case when $\mu = 2$, we show that the integral system is equivalent to the elliptic PDE system <p align="center"> -Δ $u = v^q (x)$ <br> -Δ $v = u^p (x)$ <p align="left" class="times"> in $R^n$. Our symmetry result, together with non-existence of radial solutions by Mitidieri [30], implies that, under our integrability conditions, the PDE system possesses no positive solution in the subcritical case. This partially solved the well-known Lane-Emden conjecture.

Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems

We extend the symmetry result of Gidas-Ni-Nirenberg to semilinear polyharmonic Dirichlet problems in the unit ball. In the proof we develop a new variant of the method of moving planes relying on fine estimates for the Green function of the polyharmonic operator. We also consider minimizers for subcritical higher order Sobolev embeddings. For embeddings into weighted spaces with a radially symmetric weight function, we show that the minimizers are at least axially symmetric. This result is sharp since we exhibit examples of minimizers which do not have full radial symmetry

Symmetry of non-negative solutions of a semilinear elliptic equation with singular nonlinearity

We use the method of the moving plane (MMP) to obtain necessary and sufficient conditions for the radial symmetry of positive solutions of the following semi-linear elliptic equation with singular nonlinearity: \Delta u-\frac{1}{u^\nu}=0\quad\text{in~}\mathbb{R}^n,\quad n\geq2, .