A note on the existence of positive solutions for a fourth-order semilinear elliptic system

In this note, we prove the existence of positive solutions to a class of biharmonical systems. Our main ingredients are proving a Liouville type result for biharmonical system in R＋^N via the method of moving plane combined with integral inequality, and establishing a prior estimates for positive solutions of the system via the blowing-up method.     

各项异性椭圆方程基本解的存在性

证明了右端可测的各项异性椭圆方程基本解的存在性,其中应用了各项异性Sobolev空间和Lebesgue空间.首先得到近似方程的解,然后通过对这些解的子列取极限,得到原方程的解.关键是要有一个近似函数空间以及近似方程的先验估计.最后运用Vitali定理证明了原方程基本解的存在性,推广和改进了已有方程.     

The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations

In this paper we establish the existence of a positive solution to a singular coupled system of nonlinear fractional differential equations. Our analysis rely on a nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorem in a cone.     

Existence of positive solution for a system of elliptic equations via bifurcation theory

In this paper we study the existence of solution for the following class of elliptic systems

(P)$\{\begin{array}{c}?\mathrm{\Delta }u=(a?{\int }_{\mathrm{\Omega }}K(x,y)f(u,v)dy)u+bv,\text{in}\mathrm{\Omega }\hfill \\ ?\mathrm{\Delta }v=(d?{\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(x,y)g(u,v)dy)v+cu,\text{in}\mathrm{\Omega }\hfill \\ u=v=0,\text{on}?\mathrm{\Omega }\hfill \end{array}$

where $\mathrm{\Omega }?{\mathbb{R}}^N$ is a smooth bounded domain, $N\ge 1$, and $K,\mathrm{\Gamma }:\mathrm{\Omega }\times \mathrm{\Omega }\to \mathbb{R}$ are nonnegative functions satisfying some hypotheses and $a,b,c,d\in \mathbb{R}$. The functions f and g satisfy some conditions which permit to use Bifurcation Theory to prove the existence of solution for (P).     

Proof of the favard theorem on the existence of almost-periodic solution for an arbitrary Banach space

The well-known Favard-Amerio theorem on the existence of an almost-periodic solution of a linear equation is based on the geometry of a uniformly convex space, since the almost-periodic solution is found by the minimax condition. In the present note an essentially different method for finding the almost-periodic solution is developed, which enables us to prove the Favard-Amerio theorem for an arbitrary Banach space.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 121–126, January, 1978.     

Non-existence of positive solution to a quasilinear elliptic system and blow-up estimates for a reaction-diffusion system

IntroductionIn these years, the reaction-diffusion systems of Fujita typea5 ttell as the related elliptic systemt'ith fl g RN, Tnl. n1 3 0, i = 1, 2. \vere studied by' a nu111ber of authors. The probiemsconcerning system (l) inc1ude globa1 existence and g1obal existen(.e numbers. b1ow-up. bloxv-uprates, and blow-up sets. uniqueness or nonuniqueness. et('. FOr s}'stem (2) there are problemssuch as existence or non-existence. uniqueness or nonllniqueI1ess. and so ol1. 1Ieanwhile. itseems that…     

A priori bounds and existence of positive solutions of an elliptic system of Kirchhoff type in three or four space dimensions

Let \(\Omega \subseteq \mathbb {R}^n(n\ge 3)\) be a bounded and smooth domain. Assuming that \(\alpha ,a,b>0\), \(p,q>0\), we consider the following elliptic problem of Kirchhoff type

$$\begin{aligned} \left\{ \begin{array}{llll} -(a+b\Vert \nabla u_1\Vert ^{2\alpha }_2)\Delta u_1= u_2^{p}+h_1(x, u_1, u_2, \nabla u_1, \nabla u_2) &{} \quad \text{ in }&{}\quad \Omega , \\ -\Delta u_2= u_1^{q}+h_2(x, u_1, u_2, \nabla u_1, \nabla u_2) &{}\quad \text{ in } &{}\quad \Omega , \\ u_1,u_2>0 &{} \quad \text{ in } &{}\quad \Omega , \\ u_1=u_2=0 &{} \quad \text{ on }&{}\quad \partial \Omega . \end{array} \right. \end{aligned}$$

(0.1)

Under some assumptions of \(h_i(x, u_1, u_2, \nabla u_1, \nabla u_2)(i=1, 2)\), we get a priori bounds of the positive solutions to the problem (0.1) when \(n=3,4\) and p, q satisfy \(p,q>1\), \(pq>2\alpha +1\) and \(\frac{1}{p+1}+\frac{1}{q+1}>\frac{n-2}{n}\). By these estimates and the method of continuity, we get the existence for the positive solutions to the problem (0.1). Moreover, the effect of the term \(b\Vert \nabla u_1\Vert ^{2\alpha }_2\) on the solution set of the problem (0.1) also can be given in section 2.

     

The existence, uniqueness and stability of positive periodic solution for periodic reaction-diffusion system

1. IntroductionLet fi e RN be a bounded open domain with smooth boundary afl, and for eachi E {1, 2,', m}, let Li be a second order differelltial operator defined byand Bi be a boundary operator given bywhere % denotes the outward normal derivative of m on an.We consider the following boundary value problem of the reaction-diffusion system withtime delaywhere i = 1, 2,'. I m) mr = "i(x, t -- r), r 2 0 is a constant. If r ~ 0, it means that system(I) does not include the terms of time lag…     

An existence theorem for implicit differential equations in a Banach space

Summary In this paper it is proved an existence theorem for the implicit differential equation F(t, x, x′)=0; x(t₀)=x₀ on a Banach space where it is supposed that F satisfies certain Lipschitz condition in x′ and is α-Lipschitz or α-Lipschitz in x. Entrata in Redazione il 25 maggio 1977. This research was done while the authors was visiting at the University of Minnesota.     

The existence and uniqueness theorem of the solution to a class of nonlinear fractional order system with time delay

In this paper, we investigate the existence and uniqueness of the solution to a class of nonlinear fractional order system with delay. The estimate value of the above solution is also obtained by using the generalized Gronwall inequality.     

一类弹性梁方程的正解存在性

利用Schauder不动点定理,讨论了两端简单支撑的静态梁方程 y^(4)(x)=f(x,y(x)),y(0)=a,y(1)=b,y“(0)=c,y“(1)=d在非齐边界条件下正解的存在性结果。     

The existence of a weak solution of inhomogeneous quasilinear elliptic equation with critical growth conditions

In this paper, we get the existence of a weak solution of the following inhomogeneous quasilinear elliptic equation with critical growth conditions: Supported by the Youth Foundation, Natural Science Foundation, People's Republic of China.