Steady states of radially symmetric diffusion processes

This note is concerned with the steady states of some diffusion processes arising from some physical problems. We are interested in the discussion of radially symmetric steady states, which lead to a degenerate and singular equation. Under some natural structural condition, we obtain the existence and uniqueness of nonnegative solutions.     

On the shape of solutions to an integral system related to the weighted Hardy–Littlewood–Sobolev inequality

We study the Euler–Lagrange system for a variational problem associated with the weighted Hardy–Littlewood–Sobolev inequality. We show that all the nonnegative solutions to the system are radially symmetric and have particular profiles around the origin and the infinity. This paper extends previous results obtained by other authors to the general case.     

On the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical Sobolev exponent

This paper is concerned with a semilinear parabolic equation involving critical Sobolev exponent in a ball or in RN. The asymptotic behavior of unbounded, radially symmetric, nonnegative global solutions which do not decay to zero is given. The structure of the space of initial data is also discussed.     

一类多重调和方程的正整解的存在性及其性质

该文以Schauder-Tychonoff不动点定理为工具,建立了一类R^n上带奇异性的非线性多重调和方程正整解的存在性定理,并给出了解的有关性质,所得的结果丰富和发展了已有的理论和应用。     

Symmetry of integral equation systems on bounded domains

In this paper, we investigate the symmetry of domains and solutions of integral equation systems on bounded domains. Under some natural integrability conditions, we prove that the domains are balls, all positive solutions of systems are radially symmetric and monotone decreasing with respect to the radius.     

R~2上带奇异性的非线性多重调和方程的正整解

该文以 Schauder- Tychonoff不动点定理为工具 ,建立了一类平面上带奇异性的非线性多重调和方程的正的径向对称整体解的存在性定理 ,并给出了解的有关性质 ,所得的结果丰富和发展文 [1 - 4]的结果     

Symmetry of integral equation systems with Bessel kernel on bounded domains

In this paper, we investigate the symmetry of integral equation systems with Bessel kernel on bounded domains. Under some natural integrability conditions, we prove that the domains are balls and all positive solutions are radially symmetric and monotonic decreasing.     

Uniqueness of positive radially symmetric solutions of the Dirichlet problem for a nonlinear elliptic system of second order

We prove the existence and uniqueness of positive radially symmetric solutions of the Dirichlet problem for a nonlinear elliptic system of second order. A numerical method for constructing such solutions is also given.     

Positive radially symmetric solution of the Dirichlet problem for a nonlinear elliptic system with <Emphasis Type="Italic">p</Emphasis>-Laplacian

Sufficient conditions for the existence and uniqueness of a positive radially symmetric solution of the Dirichlet problem for a nonlinear elliptic second-order system with p-Laplacian are obtained. In addition, it also proved that these conditions guarantee the nonexistence of a global positive radially symmetric solution.     

Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption

This paper deals with the Neumann initial-boundary value problem for a classical chemotaxis system with signal consumption in a disk.In contrast to previous studies which have established a comprehensive theory of global classical solutions for suitably regular nonnegative initial data,the focus in the present work is on the question to which extent initially prescribed singularities can be regularized despite the presence of the nonlinear cross-diffusive interaction.The main result in this paper asserts that at least in the framework of radial solutions immediate regularization occurs under an essentially optimal condition on the initial distribution of the population density.More precisely,it will turn out that for any radially symmetric initial data belonging to the space of regular signed Borel measures for the population density and to L² for the signal density,there exists a classical solution to the Neumann initial-boundary value problem,which is smooth and approaches the given initial data in an appropriate trace sense.     

拟线性椭圆方程的衰减正解

本文建立了一类拟线性椭圆方程具有高度衰减阶正解的存在性，并对此类正解的最大值进行了上下界估计。     

R^2上带奇异性的非线性多重调和方程的正整解

该文以Schauder-Tychonoff不动点定理为工具，建立了一类平面上带奇异性的非线性 多重调和方程的正的径向对称整体解的存在性定理，并给出了解的有关性质，所得的结果丰 富和发展文[1][4]的结果。     

Symmetry of solutions to some systems of integral equations

In this paper, we study some systems of integral equations, including those related to Hardy-Littlewood-Sobolev (HLS) inequalities. We prove that, under some integrability conditions, the positive regular solutions to the systems are radially symmetric and monotone about some point. In particular, we established the radial symmetry of the solutions to the Euler-Lagrange equations associated with the classical and weighted Hardy-Littlewood-Sobolev inequality.

     

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.     

On Harnack estimates for positive solutions of the heat equation on a complete manifold

We establish several new Harnack estimates for the nonnegative solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded by a positive or negative constant. This extends to symmetric diffusions whose generator satisfies a “curvature-dimension” inequality.     

Blow-up solutions for a general class of the second-order differential equations on the half line

In this paper, we study the existence of positive blow-up solutions for a general class of the second-order differential equations and systems, which are positive radially symmetric solutions to many elliptic problems in ${\mathbb{R}}^N$. We explore fixed point arguments applied to suitable integral equations to get solutions.     

On symmetry of nonnegative solutions of elliptic equations

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction orthogonal to H. For nonnegative solutions, this result is not always true. We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution u is symmetric about H  . Moreover, we prove that if u?0$u?0$, then the nodal set of u divides the domain Ω into a finite number of reflectionally symmetric subdomains in which u has the usual Gidas–Ni–Nirenberg symmetry and monotonicity properties. We also show several examples of nonnegative solutions with a nonempty interior nodal set.     

一类二阶非线性方程奇异边值问题

本文证明了方程div（|Du|p-2Du） f(r.u（r），u'(r))=0（R1＜r＜R0）正对称解的存在性．这里f允许在u=0或。u’=0处奇异．     

Positive entire solutions for a class of singular nonlinear polyharmonic equations on

In the paper, the existence of positive radially symmetric entire solutions for a class of singular nonlinear polyharmonic equations on is proved. Some properties are obtained. The results of this paper are generalizations of those in Refs. [Math. Z. 67 (1957) 32; Arch. Math. 9 (1958) 308; Proc. R. Soc. Edinburg 82A (1979) 189; Hiroshima Math. J. 17 (1989) 13].     

Symmetry results for decay solutions of elliptic systems in the whole space

In this paper, by using the Alexandrov-Serrin method of moving planes combined with maximum principles, we prove that the decaying positive solutions of a semi-linear elliptic system in the whole space are radially symmetric about some point. The system under our consideration includes the important physical interesting case, the stationary Schrödinger system for Bose-Einstein condensate.