Structure of Positive Solutions of Quasilinear Elliptic Equations with Critical and Supercritical Growth

The structure of positive radial solutions to a class of quasilinear elliptic equations with critical and supercritical growth is precisely studied. A large solution and a small solution are obtained for the equations. It is shown that the large solution is unique, its asymptotic behaviour and flat core are also discussed.     

On Uniqueness of Boundary Blow-Up Solutions of a Class of Nonlinear Elliptic Equations

We study boundary blow-up solutions of semilinear elliptic equations Lu = u ₊ ^p with p > 1, or Lu = e ^au with a > 0, where L is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.     

Morrey Regularity of Solutions to Degenerate Elliptic Equations in Rn

In this paper, we study the Morrey regularity of solutions to the de- generate elliptic equation -(a_{ij}u_{xi})_{xj} = -(f_j)_{xj} in R^n. For this purpose, we introduce four weighted Morrey spaces in R^n.     

半线性椭圆方程的正解

本文用特征理论及上下解方法,证明了一类半线性椭圆方程边值问题的正解的存在性,同时给出了解的估计.     

一类次线性椭圆方程组正解的存在唯一性

考虑半线性椭圆方程组■(1)其中A>0,Ω是有界光滑区域.f,g是定义在R_+~2:=[0,∞)×[0,∞)上的实值函数讨论在满足什么条件下此半线性椭圆方程组存在唯一的正解.     

Asymptotics for Solutions of Elliptic Equations in Double Divergence Form

We consider weak solutions of an elliptic equation of the form ? _i ? _i (a _ij u) = 0 and their asymptotic properties at an interior point. We assume that the coefficients are bounded, measurable, complex-valued functions that stabilize as x → 0 in that the norm of the matrix (a _ij (x) ? δ _ij ) on the annulus B _2r \ B _r is bounded by a function Ω(r), where Ω²(r) satisfies the Dini condition at r = 0, as well as some technical monotonicity conditions; under these assumptions, solutions need not be continuous. Our main result is an explicit formula for the leading asymptotic term for solutions with at most a mild singularity at x = 0. As a consequence, we obtain upper and lower estimates for the L ^p -norm of solutions, as well as necessary and sufficient conditions for solutions to be bounded or tend to zero in L ^p -mean as r → 0.     

General Uniqueness Results and Variation Speed for Blow-Up Solutions of Elliptic Equations

Let be a smooth bounded domain in R^N. We prove general uniquenessresults for equations of the form – u = au – b(x)f(u) in , subject to u = on . Our uniqueness theorem is establishedin a setting involving Karamata's theory on regularly varyingfunctions, which is used to relate the blow-up behavior of u(x)with f(u) and b(x), where b 0 on and a certain ratio involvingb is bounded near . A key step in our proof of uniqueness usesa modification of an iteration technique due to Safonov. 2000Mathematics Subject Classification 35J25 (primary), 35B40, 35J60(secondary).     

全空间上一类半线性椭圆方程的正解

本文研究Rn(n≥ 3)上的半线性椭圆方程 -Δu=f(u)的C2 正解 .如果f(u)在 ( 0 ,∞ )上局部有界并且对某个α 1 <α相似文献   

退化的非线性椭圆方程的径向解

利用打靶法讨论一类退化非线性椭圆型方程径向解的性态，得出了径向解的间断及不间断的结果．     

一类非线性椭圆方程解的存在性

本文考虑一类非线性椭圆方程,运用强制变分方法,我们给出了在多种情况下方程解的存在性。     

Asymptotics of Solutions for Nonlocal Elliptic Problems in Plane Angles

In the paper we investigate asymptotics of solutions for nonlocal elliptic problems in plane angles and in ² \ {0}. These problems arise in studying nonlocal problems in bounded domains in the case where support of nonlocal terms intersects with a boundary of a domain. We obtain explicit formulas for the asymptotic coefficients in terms of eigenvectors and associated vectors of both adjoint nonlocal operators acting in spaces of distributions and formally adjoint (with respect to the Green formula) nonlocal transmission problems.     

一类奇异临界椭圆方程非平凡解的存在性

研究了一类带Sobolev-Hardy临界指数的奇异椭圆方程,应用变分方法,通过能量估计和证明对应的能量泛函满足(PS)_c条件,运用山路引理得到了这类方程非平凡解的存在性.     

Unique Continuation Property and Local Asymptotics of Solutions to Fractional Elliptic Equations

Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied in this paper. By using an Almgren type monotonicity formula, separation of variables, and blow-up arguments, we describe the exact behavior near the singularity of solutions to linear and semilinear fractional elliptic equations with a homogeneous singular potential related to the fractional Hardy inequality. As a consequence we obtain unique continuation properties for fractional elliptic equations.     

On the Interior Smoothness of Solutions to Second-Order Elliptic Equations

We study the interior smoothness properties of solutions to a linear second-order uniformly elliptic equation in selfadjoint form without lower-order terms and with measurable bounded coefficients. In terms of membership in a special function space we combine and supplement some properties of solutions such as membership in the Sobolev space W _{2, loc} ¹ and Holder continuity. We show that the membership of solutions in the introduced space which we establish in this article gives some new properties that do not follow from Holder continuity and the membership in W _2,loc ¹ .     

拟线性椭圆型方程带有混合奇异项的正整体解

在本文中,研究了方程div(｜u｜p-2u) f(x,u)=0,x∈RN,N≥3的正整体解,其中f(x,u)在u=0未假定是正则的,且f(x,u)可以同时包含超线性,亚线性项和奇异项.     

Integrability of Solutions of Nonlinear Elliptic Equations with Right-Hand Sides from Classes Close to L 1

We establish a number of results on the integrability of the entropy and the weak solutions of the Dirichlet problem for nonlinear elliptic equations of second order depending on the integrability properties of the right-hand sides of these equations.     

Higher Integrability of the Gradient in Degenerate Elliptic Equations

We prove that under some global conditions on the maximum and the minimum eigenvalue of the matrix of the coefficients, the gradient of the (weak) solution of some degenerate elliptic equations has higher integrability than expected. Technically we adapt the Giaquinta–Modica regularity method in some degenerate cases. When the dimension is two, a consequence of our result is a new Hölder continuity result for the weak solution.     

Irregular Solutions of Linear Degenerate Elliptic Equations

We give examples of discontinuous solutions and of unbounded solutions of linear isotropic degenerate elliptic equations. Discontinuous solutions exist even when both the maximum eigenvalue and the inverse of the minimum eigenvalue of the matrix of the coefficients are in the intersection of all the Lp spaces.     

Non-Uniformly Elliptic Equations with Natural Growth in the Gradient

We prove the existence of bounded solutions for a class of nonlinear elliptic problems of type–div(a(x,u,Du))=H(x,u,Du)+f, uW ^1,p ₀()L (),where a(x,,)b(||)|| ^p , b is a continuous monotone decreasing function and |H(x,,)| k()|| ^p , k is a continuous monotone increasing function.     

Multiple Solutions for a Class of Semilinear Elliptic Equations

We show that for a class of semilinear elliptic equations there are at least three nontrivial solutions. Existence of infinitely many solutions is also shown when the nonlinear term is odd. In our results, the nonlinear term can grow super-critically at infinity.