带有临界指数的椭圆方程多重正解的存在性

采用变分方法考虑了一类带有临界指数和有限个奇点的椭圆方程正解的存在性问题,利用临界点理论可以得到V(x)的每一个奇点都可以产生一个正解.     

奇异椭圆方程多重解的存在性

在这篇论文中,首先给出了奇异椭圆方程(1.1)正解在零点附近的一个精确的估计.然后,结合这个估计式,利用Ekeland变分原理和山路引理,在一定条件下得到了方程(1.1)多重正解的存在性.     

一类含临界耦合非线性项的奇异椭圆方程组的正解

本文研究了一类含临界指数与耦合非线性项的奇异椭圆方程组. 利用变分方法与极大值原理, 通过证明对应的能量泛函满足局部的 (PS)c 条件, 得到了这类方程组正解的存在性, 推广了单个方程与方程组中的相应结果.     

Critical points and positive solutions of singular elliptic boundary value problems

Usually we do not think there is variational structure for singular elliptic boundary value problems, so it cannot be considered by using critical points theory. In this paper, we use critical theory on certain convex closed sets to solve positive solutions for singular elliptic boundary value problems, especially use the ordinary differential equation theory of Banach spaces to obtain new results on the existence of multiple positive solutions. The method is useful for other singular problems.     

Some further results on a semilinear equation with concave–convex nonlinearity

Following our previous work [J. Math. Anal. Appl. 295 (2004) 341], we give an exact growth order near zero for positive solutions of a class of elliptic equation and use it to give the existence of multiple solutions with negative energy, multiple positive solutions and sign changing solutions of the considered problem. Even in the particular case, our results extend a previous work by Ambrosetti–Garcia–Peral [J. Funct. Anal. 137 (1996) 219].     

Sobolev type embedding and weak solutions with a prescribed singular set

New Sobolev type embeddings for some weighted Banach spaces are established. Using such embeddings and the singular positive radial entire solutions, we construct singular positive weak solutions with a prescribed singular set for a weighted elliptic equation. Our main results in this paper also provide positive weak solutions with a prescribed singular set to an equation with Hardy potential.     

Positive solutions of a nonlocal singular elliptic equation by means of a non-standard bifurcation theory

We consider a nonlocal elliptic equation arising in a prey–predator model whose nonlocal term is singular. We use the Leray–Schauder degree to prove the existence of an unbounded continuum of positive solutions emanating from the trivial solution. As application, we study nonlocal and singular elliptic equations of the type logistic and Holling–Tanner.     

ASYMPTOTICS FOR POSITIVE SOLUTIONS OF THE WEIGHTED LAPLACE EQUATIONS INVOLVING CRITICAL GROWTH

1IntroductionConsiderthefollowingproblemwl1ereflCR",n23,isaboundeddomainwithsmoothboundarycontainingtheorigin.TheexponentscrjP,rrandpsatisfyItisknownthatthefollowillgS(i1)'llf'v-Hardyinequalityholdsifandonlyif(1.2)issatisfied(cL[4])foralluECj(R"),theconstantS(cr,P)seebelow(1.5).Weshallbeconcernedwiththecasesp5o,p 1>2andcrta>P-2.DenotebyHt(fl)thespaceofcompletionofCi(fl)undertheinnerproduct(u,v)=jnlxld7u'7l)dx.uEH:(fl)iscalledaweaksolutionof(1-1)iff0rallvEHt(fl).Ithasbeeushownin[5]thata…     

Exact local behavior of positive solutions for a semilinear elliptic equation with Hardy term

We characterize an exact growth order near zero for positive solutions of a semilinear elliptic equation with Hardy term. This result strengthens an existence result due to E. Jannelli [The role played by space dimension in elliptic critical problems, JDE 156 (1999), 407-426].

     

Multiple positive solutions for a semilinear equation with prescribed singularity

Variational methods are used to prove the existence of multiple positive solutions for a semilinear equation with prescribed finitely many singular points. Some exact local behavior for positive solutions are also given.     

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.     

Multiple positive solutions for a class of nonlinear elliptic equations

Via delicate estimates, we characterize an exact growth order near zero for positive solutions of a class of nonlinear elliptic equations. Using this characterization, we obtain multiple positive solutions for equations involving critical nonlinearity.     

Properties of positive solutions to an elliptic equation with negative exponent

In this paper, we study some quantitative properties of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics as well as differential geometry. We can get some useful local gradient estimate and L¹ lower bound for positive solutions of the elliptic equation. A uniform positive lower bound for convex positive solutions is also obtained. We show that in lower dimensions, there is no stable positive solutions in the whole space. In the whole space of dimension two, we can show that there is no positive smooth solution with finite Morse index. Symmetry properties of related integral equations are also given.     

Graded mesh refinement and error estimates of higher order for DGFE solutions of elliptic boundary value problems in polygons

Error estimates for DGFE solutions are well investigated if one assumes that the exact solution is sufficiently regular. In this article, we consider a Dirichlet and a mixed boundary value problem for a linear elliptic equation in a polygon. It is well known that the first derivatives of the solutions develop singularities near reentrant corner points or points where the boundary conditions change. On the basis of the regularity results formulated in Sobolev–Slobodetskii spaces and weighted spaces of Kondratiev type, we prove error estimates of higher order for DGFE solutions using a suitable graded mesh refinement near boundary singular points. The main tools are as follows: regularity investigation for the exact solution relying on general results for elliptic boundary value problems, error analysis for the interpolation in Sobolev–Slobodetskii spaces, and error estimates for DGFE solutions on special graded refined meshes combined with estimates in weighted Sobolev spaces. Our main result is that there exist a local grading of the mesh and a piecewise interpolation by polynoms of higher degree such that we will get the same order O (h^α) of approximation as in the smooth case. © 2011 Wiley Periodicals, Inc. Numer Mehods Partial Differential Eq, 2012     

Elliptic function solutions for some nonlinear pdes in mathematical physics

In this work, we have constructed various types of soliton solutions of the generalized regularized long wave and generalized nonlinear Klein-Gordon equations by the using of the extended trial equation method. Some of the obtained exact traveling wave solutions to these nonlinear problems are the rational function, 1-soliton, singular, the elliptic integral functions $F, E, \Pi$ and the Jacobi elliptic function sn solutions. Also, all of the solutions are compared with the exact solutions in literature, and it is seen that some of the solutions computed in this paper are new wave solutions.     

MULTIPLICITY OF POSITIVE SOLUTIONS FOR SINGULAR ELLIPTIC SYSTEMS WITH CRITICAL SOBOLEV-HARDY AND CONCAVE EXPONENTS

In this paper,we consider a singular elliptic system with both concave non-linearities and critical Sobolev-Hardy growth terms in bounded domains.By means of variational methods,the multiplicity of positive solutions to this problem is obtained.     

Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent

We studied the asymptotic behavior of local solutions for strongly coupled critical elliptic systems near an isolated singularity. For the dimension less than or equal to five we prove that any singular solution is asymptotic to a rotationally symmetric Fowler type solution. This result generalizes the celebrated work due to Caffarelli, Gidas and Spruck [1] who studied asymptotic proprieties to the classic Yamabe equation. In addition, we generalize similar results by Marques [12] for inhomogeneous context, that is, when the metric is not necessarily conformally flat.     

On a nonlinear elliptic problem with critical potential in ?2

Consider the existence of nontrivial solutions of homogeneous Dirichlet problem for a nonlinear elliptic equation with the critical potential in ℝ². By establishing a weighted inequality with the best constant, determine the critical potential in ℝ², and study the eigenvalues of Laplace equation with the critical potential. By the Pohozaev identity of a solution with a singular point and the Cauchy-Kovalevskaya theorem, obtain the nonexistence result of solutions with singular points to the nonlinear elliptic equation. Moreover, for the same problem, the existence results of multiple solutions are proved by the mountain pass theorem.     

一类带奇异系数椭圆方程解的存在性

本文运用变分方法及Hardy不等式讨论了一类带奇异系数的临界椭圆方程，证明了在一定条件下方程解的存在性。