Uniqueness of a positive solution to the Dirichlet problem for a quasilinear equation with <Emphasis Type="Italic">p</Emphasis>-laplacian in a ball

We obtain sufficient conditions for the existence of a unique positive radially symmetric solution to the Dirichlet problem for a quasilinear equation of elliptic type in a multidimensional ball.     

Radially Symmetric Solutions of the <Emphasis Type="Italic">p</Emphasis>-Laplace Equation with Gradient Terms

We consider the Dirichlet problem for the p-Laplace equation with nonlinear gradient terms. In particular, these gradient terms cannot satisfy the Bernstein—Nagumo conditions. We obtain some sufficient conditions that guarantee the existence of a global bounded radially symmetric solution without any restrictions on the growth of the gradient term. Also we present some conditions on the function simulating the mass forces, which allow us to obtain a bounded radially symmetric solution under presence of an arbitrary nonlinear source.     

Nonradial positive solutions of the ‐Laplace Emden–Fowler equation with sign‐changing weight

In this paper we study the p‐Laplace Emden–Fowler equation with a radial and sign‐changing weight in the unit ball under the Dirichlet boundary condition. We show that if the weight function is negative in the unit ball except for a small neighborhood of the boundary and positive at somewhere in this neighborhood, then no least energy solution is radially symmetric. Therefore the equation has both a positive radial solution and a positive nonradial solution. Moreover, we prove in the one dimensional case that if the neighborhood is large, then a positive solution is unique.     

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 ? ^N  × (?∞, T) decaying to zero at spatial infinity uniformly with respect to time is radially symmetric around some origin in ? ^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.     

A comparison result for perturbed radial p-Laplacians

Consider the radially symmetric p-Laplacian for p?2 under zero Dirichlet boundary conditions. The main result of the present paper is that under appropriate conditions a solution of a perturbed (radially symmetric) p-Laplacian can be compared with the solution of the unperturbed one. As a consequence one obtains a sign preserving result for a system of p-Laplacians which are coupled in a nonquasimonotone way.     

Nontopological N‐vortex condensates for the self‐dual Chern‐Simons theory

We prove the existence of nontopological N‐vortex solutions for an arbitrary number N of vortex points for the self‐dual Chern‐Simons‐Higgs theory with 't Hooft “periodic” boundary conditions. We use a shadowing‐type lemma to glue together any number of single vortices obtained as a perturbation of a radially symmetric entire solution of the Liouville equation. © 2003 Wiley Periodicals, Inc.     

EXISTENCE OF POSITIVE RADIAL SOLUTIONS FOR QUASILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

1IntroductionInrecentyears,theexistenceandmultiplicityofsolutionsoftheproblemwerestudiedbynumerousauthors.WhereA>0,fiisasmoothdomaininR",N22.Thefundamentalquestionsaboveproblem(1.1)-(1.2):have.receivedrathersimpleanswers.Todoso,theauthorsalwaysusethenicepropertiesoftheoperatorA,forexample,thelinearity,maximumprinciple,strongmaximumprinciple,comparisonprincipleandsoon.Meanwhile,theexistenceandmultiplicityofpositiveradialsolutionsof(1.1)-(1.2)attractedmuchinterest,see[l,5-6,10,13-18]andtherefe…     

Existence of radially symmetric solutions of the inhomogeneous <Emphasis Type="Italic">p</Emphasis>-Laplace equation

We consider the Dirichlet problem for the inhomogeneous p-Laplace equation with p nonlinear source. New sufficient conditions are established for the existence of weak bounded radially symmetric solutions as well as a priori estimates of solution and of the gradient of solution. We obtain an explicit formula that shows the dependence of the existence of these solutions on the dimension of the problem, the size of the domain, the exponent p, the nonlinear source, and the exterior mass forces.     

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

We consider quasilinear parabolic equations on ? ^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.     

Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive numbers ρ<R, there exists a radially symmetric stationary solution with tumor boundary r=R and necrotic core boundary r=ρ. The system depends on a positive parameter μ, which describes the tumor aggressiveness. There also exists a sequence of values μ₂<μ₃<? for which branches of symmetry-breaking stationary solutions bifurcate from the radially symmetric solution branch.     

Radially symmetric growth of nonnecrotic tumors

The growth of tumors is an important subject in recent research. We present here a mathematical model for the growth of nonnecrotic tumors in all the three regimes of vascularisation. This leads to a free-boundary problem which we treat by means ODE techniques. We prove the existence of a unique radially symmetric stationary solution. It is also shown that, if the initial tumor is radially symmetric, there exists a unique radially symmetric solution of the evolution equation, which exists for all times. The asymptotic behaviour of this solution will be discussed in relation to the parameters characterizing cell proliferation and cell death.     

Global existence of the radially symmetric solutions of the Navier–Stokes equations for the isentropic compressible fluids

We study the isentropic compressible Navier–Stokes equations with radially symmetric data in an annular domain. We first prove the global existence and regularity results on the radially symmetric weak solutions with non‐negative bounded densities. Then we prove the global existence of radially symmetric strong solutions when the initial data ρ₀, u ₀ satisfy the compatibility condition for some radially symmetric g ∈ L². The initial density ρ₀ needs not be positive. We also prove some uniqueness results on the strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

一类$R^{N}(N\geq 3)$上奇异非线性双调和方程正整解

建立了RN(N≥3)上一类奇异非线性双调和方程正的径向对称整体解的存在定理,并给出了解的有关性质,推广了相关文献的结果.     

Existence and multiplicity of positive solutions to Schrödinger–Poisson type systems with critical nonlocal term

The existence, nonexistence and multiplicity of positive radially symmetric solutions to a class of Schrödinger–Poisson type systems with critical nonlocal term are studied with variational methods. The existence of both the ground state solution and mountain pass type solutions are proved. It is shown that the parameter ranges of existence and nonexistence of positive solutions for the critical nonlocal case are completely different from the ones for the subcritical nonlocal system.     

Radially symmetric convex solutions for Dirichlet problems of Monge‐Ampère equations

In this paper, we study the existence of radially symmetric convex solutions for Dirichlet problems of Monge‐Ampère equations. By applying a well‐known fixed point theorem in cones, we shall establish several new criteria for the existence of nontrivial radially symmetric convex solutions for the systems of Monge‐Ampère equations with or without an eigenvalue parameter. Copyright © 2015 John Wiley & Sons, Ltd.     

Shell dynamics in the presence of a central body

We construct an idealized spherically symmetric relativistic model of an exploding object in the framework of the theory of surface layers in general relativity and match a Vaidya solution for a radially radiating star to another Vaidya solution through a thin spherical shell. We reduce the equations of motion and the radiation density of the Vaidya solution given by the matching conditions to a first-order system and analyze the general characteristics of the motion. We use a post-Newtonian approximation to find the equation of motion of a spherically symmetric radiating shell moving in a central gravitational potential.     

The blowup mechanism for 3-D quasilinear wave equations with small data

For a class of special three-dimensional quasilinear wave equations, we study the blowup mechanism of classical solutions. More precisely, under the nondegenerate conditions, any radially symmetric solution with small initial data is shown to develop singularities in the second order derivatives while the first order derivatives and itself remain continuous, moreover the blowup of solution is of “cusp type”.     

Nonnegative solutions to a semilinear dirichlet problem in a ball are positive and radially syhhetric*

We prove that nonnegative.so1utions to a semilinear Dirichlet problem in a ball are positive, and hence radially symmetric. In particular this answers a question in [3] where positive solutions were proven to be radially symmetric. In section 4 we provide a sufficient condition on the geometry of the .domain which ensures that nonnegative solutions are positive in the interior.     

Radially symmetric minimizers for a p-Ginzburg Landau type energy in {\mathbb R^2}

We consider the minimization of a p-Ginzburg-Landau energy functional over the class of radially symmetric functions of degree one. We prove the existence of a unique minimizer in this class, and show that its modulus is monotone increasing and concave. We also study the asymptotic limit of the minimizers as p ?? ??. Finally, we prove that the radially symmetric solution is locally stable for 2?<?p????4.     

A symmetry result for the p-laplacian equation via the moving planes method

Thsi paper is concerned with a positive solution u of the non–homogeneous p–Laplacian equation in an open, bounded, connected subset Ω of Rⁿ with C² boundary. We assume that u verifies overdetermined boundary conditions and we prove that of us has only one critial point Ω thenΩ is a ball and u is radially symmetric; to prove this result we use the moving planes method introduced by J.Serrien. Using the same technique we also prove that the result is stable in the following sense: the boundary of Ω tends to the boundary of a sphere as the diameter of the critical set u tends to 0.