Symmetry Properties Of Positive Solutions Research Articles

Symmetry of solutions for asymptotically symmetric nonlocal parabolic equations

In this paper, we consider the symmetry properties of positive solutions for nonlocal parabolic equations in the whole space. We obtain various asymptotic maximum principles for carrying out the asymptotic method of moving planes. With the help of these results, we show that if the equation converges to a symmetric one, then the solutions will converge to radially symmetric functions. The methods and techniques used here can be easily applied to study a variety of nonlocal parabolic equations with more general operators and nonlinear terms.

Symmetry properties of positive solutions for fully nonlinear elliptic systems

We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such asFi(x,Dui,D2ui)+fi(x,u1,…,un,Dui)=0,1≤i≤n, in a bounded domain Ω in RN with Dirichlet boundary condition u1=…,un=0 on ∂Ω. Here, fi's are nonincreasing with the radius r=|x|, and satisfy a cooperativity assumption. In addition, each fi is the sum of a locally Lipschitz with a nondecreasing function in the variable ui, and may have superlinear gradient growth. We show that symmetry occurs for systems with nondifferentiable fi's by developing a unified treatment of the classical moving planes method in the spirit of Gidas-Ni-Nirenberg. We also present different applications of our results, including uniqueness of positive solutions for Lane-Emden systems in the subcritical case in a ball, and symmetry for a class of systems with natural growth in the gradient.

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.

SYMMETRY AND EXISTENCE OF SOLUTIONS OF SEMI-LINEAR ELLIPTIC SYSTEMS IN HYPERBOLIC SPACE

Abstract(0.1) \begin{equation}\label{eq:0.1} \left\{ \begin{array}{ll} \displaystyle -\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v x, \\ \displaystyle -\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, \\ \end{array} \right. \end{equation} in the whole Hyperbolic space ℍN. We establish decay estimates and symmetry properties of positive solutions. Unlike the corresponding problem in Euclidean space ℝN, we prove that there is a positive solution pair (u, v) ∈ H1(ℍN) × H1(ℍN) of problem (0.1), moreover a ground state solution is obtained. Furthermore, we also prove that the above problem has a radial positive solution.

Symmetry Properties of Positive Solutions of Parabolic Equations on ℝ N : II. Entire Solutions

We consider nonautonomous quasilinear parabolic equations satisfying certain symmetry conditions. We prove that each positive bounded solution u on ℝ N × (−∞, T) decaying to zero at spatial infinity uniformly with respect to time is radially symmetric around some origin in ℝ N . The origin depends on the solution but is independent of time. We also consider the linearized equation along u and prove that each bounded (positive or not) solution is a linear combination of a radially symmetric solution and (nonsymmetric) spatial derivatives of u. Theorems on reflectional symmetry are also given.

Symmetry Properties of Positive Solutions of Parabolic Equations on ℝ N : I. Asymptotic Symmetry for the Cauchy Problem

We consider quasilinear parabolic equations on ℝ N satisfying certain symmetry conditions. We prove that bounded positive solutions decaying to zero at spatial infinity are asymptotically radially symmetric about a center. The asymptotic center of symmetry is not fixed a priori (and depends on the solution) but it is independent of time. We also prove a similar theorem on reflectional symmetry.

Radial Symmetry of Self-Similar Solutions for Semilinear Heat Equations

The symmetry properties of positive solutions of the equationΔu+12x·∇u+1p−1u+up=0inRn,where n⩾2, p>(n+2)/n, was studied. It was proved that u must be radially symmetric about the origin provided u(x)=o(|x|−2/(p−1)) as |x|→∞, and that there exist non-radial solutions u satisfying limsup|x|→∞|x|2/(p−1)u(x)>0.

Monotonicity and symmetry results for $p$-Laplace equations and applications

In this paper we prove monotonicity and symmetry properties of positive solutions of the equation $ - div (|Du|^{p-2}Du)=f(u)$, $1 <p <2$, in a smooth bounded domain ${\Omega} $ satisfying the boundary condition $u=0$ on ${\partial} {\Omega} $. We assume $f$ locally Lipschitz continuous only in $(0, \infty) $ and either $f \geq 0 $ in $[0, \infty] $ or $f$ satisfying a growth condition near zero. In particular we can treat the case of $f(s) = s^{{\alpha}} -c\, s^q $, ${\alpha} >0 $, $ c \geq 0 $, $ q \geq p-1 $. As a consequence we get an extension to the $p$--Laplacian case of a symmetry theorem of Serrin for an overdetermined problem in bounded domains. Finally we apply the results obtained to the problem of finding the best constants for the classical isoperimetric inequality and for some Sobolev embeddings.

A note on radial symmetry of positive solutions for semilinear elliptic equations in $\mathbb{R}^n$

Symmetry properties of positive solutions of the equations $$ \Delta u + \phi(|x|)f(u) = 0 $$ in $\mathbb{R}^n$ are considered. We employ the moving plane method based on the maximum principle on unbounded domains to obtain new results on symmetr

Symmetry properties of positive solutions of elliptic equations on symmetric domains

An improved version of the maximum principle is used to generalize Alexandrov's moving-planes method. Symmetry and monotonicity properties are obtained for positive solutions of second order semilinear elliptic equations in symmetric domains in ℝn as well as in domains in Riemannian manifolds.

Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions

We study symmetry properties of positive solutions to some semilinear elliptic problems with nonlinear Neumann boundary conditions. We give sufficient conditions to have symmetry around the $\e_n$-axis of positive solutions of problems on the half-space. The proofs are based on the moving plane method. Finally some symmetry results are given in the case when the domain is a ball.

Symmetrization properties of parabolic equations in symmetric domains

Symmetry properties of positive solutions of a Dirichlet problem for a strongly nonlinear parabolic partial differential equation in a symmetric domainD ⊂ R n are considered. It is assumed that the domainD and the equation are invariant with respect to a group {Q} of transformations ofD. In examples {Q} consists of reflections or rotations. The main result of the paper is the theorem which states that any compact inC(D) negatively invariant set which consists of positive functions consists ofQ-symmetric functions. Examples of negatively invariant sets are (in autonomous case) equilibrium points, omega-limit sets, alpha-limit sets, unstable sets of invariant sets, and global attractors. Application of the main theorem to equilibrium points gives the Gidas-Ni-Nirenberg theorem. Applying the theorem to omega-limit sets, we obtain the asymptotical symmetrization property. That means that a bounded solutionu(t) asr→∞ approaches subspace of symmetric functions. One more result concerns properties of eigenfunctions of linearizations of the equations at positive equilibrium points. It is proved that all unstable eigenfunctions are symmetric.

Symmetry properties of positive solutions of elliptic equations in an infinite sectorial cone

SynopsisWe prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.