Multiplicity of Solutions for a Class of Noncooperative Elliptic Systems

In this paper, we consider the following noncooperative elliptic systems where Ω is a bounded domain in RN with smooth boundary ∂Ω, λ,δ,γ are real parameters, and . We assume that F is subquadratic at zero with respect to the variables u,v. By using a variant Clark’s theorem, we obtain infinitely many nontrivial solutions (uk,vk) with as k → ∞. Compared with the existing literature, we do not need to assume the behavior of the nonlinearity ∇F at infinity.

OPTIMAL INTERIOR PARTIAL REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS UNDER THE NATURAL GROWTH CONDITION:THE METHOD OF A-HARMONIC APPROXIMATION

In this article, the authors consider the nonlinear elliptic systems under the natural growth condition. They use a new method introduced by Duzaar and Grotowski, for proving partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation. And directly establish the optimal Holder exponent for the derivative of a weak solution.

OPTIMAL INTERIOR PARTIAL REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS WITH DINI CONTINUOUS COEFFICIENTS

In this article, we consider nonlinear elliptic systems of divergence type with Dini continuous coefficients. The authors use a new method introduced by Duzaar and Grotowski, to prove partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation and directly establish the optimal HSlder exponent for the derivative of a weak solution on its regular set.

MULTIPLE SYMMETRIC RESULTS FOR A CLASS OF BIHARMONIC ELLIPTIC SYSTEMS WITH CRITICAL HOMOGENEOUS NONLINEARITY IN R^N

This paper is dedicated to studying the existence and multiplicity of symmetric solutions for a class of biharmonic elliptic systems with critical homogeneous nonlinearity in RN. By virtue of variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the parameters and the weighted functions.

INFINITELY MANY SOLUTIONS FOR ELLIPTIC SYSTEMS WITH STRONGLY INDEFINITE VARIATIONAL STRUCTURE

Based on the multiplicity results of Benci and Fortunato [4], we consider some elliptic systems with strongly indefinite quadratic part, and establish the existence of infinitely many nontrivial solutions in a suitable family of products of fractional Sobolev spaces.

Nonlinear Riemann Problem for Nonlinear Elliptic Systems in Sobolev Space W_(1,p)(D)

The nonlinear Riemann problems were converted into nonlinear singular integral equ ations and the existence of the solution for the problem was proved by means of contract principle.

C^(1,α)-PARTIAL REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS

We prove C1.α-partial regularity of weak solutions of nonlinear elliptic systems under the main assumption that Aia and Bi satisfy the controllable growth condition or the natural growth condition.

A CLASS OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR THE SECOND-ORDER E_2 CLASS ELLIPTIC SYSTEMS IN GENERAL FORM

A class of nonlinear boundary value problems(BVP) for the second_order E 2 class elliptic systems in general form is discussed. By introducing a kind of transformation,this kind of BVP is reduced to a class of generalized nonlinear Riemann_Hilbert BVP. And then some singular integral operators are introduced to establish the equivalent nonlinear singular integral equations. The solvability is proved under some suitable hypotheses by means of the properties of singular integral operators and the function theoretic methods.

REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS WITH DINI COEFFICIENTS UNDER NATURAL GROWTH CONDITION FOR THE CASE:1

In this article,we consider interior regularity for weak solutions to nonlinear elliptic systems of divergence type with Dini continuous coefficients under natural growth condition for the case 1〈m〈 2.All estimates in the case of m≥2 is no longer suitable,and we can’t obtain the Caccioppoli’s second inequality by using these techniques developed in the case of m≥2.But the Caccioppoli’s second inequality is the key to use A-harmonic approximation method.Thus,we adopt another technique introduced by Acerbi and Fcsco to overcome the difficulty and we also overcome those difficulties due to Dini condition.And then we apply the A-harmonic approximation method to prove partial regularity of weak solutions.

REGULARITY FOR CERTAIN QUASILINEAR ELLIPTIC SYSTEMS OF DIVERGENCE STRUCTURE

The regularity of the gradient of Holder continuous solutions of quasi-linear elliptic systems of the form -Dj(aij(x,u,Du)Diuk)=-Difki=gk is investigated. Partial regularity and ε-regularity are shown to hold under the structural assumption -Dj(aij(x,u,Du))=hi∈L∞

REGULARITY OF H^1 ∩L^(n((γ-1)/(2-γ))) WEAK SOLUTIONS FOR NONLINEAR ELLIPTIC SYSTEMS

Let us consider the following elliptic systems of second order-D_α(A_i~α(x, u, Du))=B_4(x, u, Du), i=1, …, N, x∈Q(?)R^n, n≥3 (1) and supposeⅰ) |A_i~α(x, u, Du)|≤L(1+|Du|);ⅱ) (1+|p|)^(-1)A_i~α(x, u, p)are H(?)lder-continuous functions with some exponent δ on (?)×R^N uniformly with respect to p, i.e.ⅲ) A_i~α(x, u, p) are differentiable function in p with bounded and continuous derivativesⅳ)ⅴ) for all u∈H_(loc)~1(Ω, R^N)∩L^(n(γ-1)/(2-γ))(Ω, R^N), B(x, u, Du)is ineasurable and |B(x, u, p)|≤a(|p|~γ+|u|~τ)+b(x), where 1+2/n<γ<2, τ≤max((n+2)/(n-2), (γ-1)/(2-γ)-ε), (?)ε>0, b(x)∈L2n/(n+2), n^2/(n+2)+e(Ω), (?)ε>0.Remarks. Only bounded open set Q will be considered in this paper; for all p≥1, λ≥0, which is clled a Morrey Space.Let assumptions ⅰ)-ⅳ) hold, Giaquinta and Modica have proved the regularity of both the H^1 weak solutions of (1) under controllable growth condition |B|≤α(|p|~γ+|u|^((n+2)/(n-2))+b, 0<γ≤1+2/n and the H^1∩L~∞ weak solutions of (1) under natural growth condition |B|≤α|p|~2+b with a smallness condition 2aM<λ(|u|≤M), which implys that the H^1∩L~∞ weak solutions have the same regularty in the case of 1+2/n<γ<2. In the case of γ=2, many counterexamples (see [2] showed that u must be in H^1L~∞, while in the case of 1+2/n<γ<2, we consider the H^1∩L^n(γ-1)/(2-γ) weak solutions of (1), weaken the instability conditions upon them (from L~∞ to L^n(γ-1)/(2-γ) and obtain the same regularity results. Finally we show that the exponent n(γ-1)/(2-γ) can not be docreased anymore for the sake of the regularity results.Delinition 1. We call u∈H^1∩L^n(γ-1)/(2-γ)(Q, R^N) be a weak solution of (1), providod that where We use the convention that repeated indices are summed. i, j go from 1 to N ann α, β from 1 to n.

ON THE EIGENVALUE PROBLEM FOR ELLIPTIC SYSTEMS WITH STRONGLY EIGEN-EXPONENT UNDER NATURAL GROWTH CONDITIONS

Let Ω be a bounded domain with smooth boundary Ω in R~n. We consider the following eigenvalue problem for systems of elliptic equations under the natural growth conditions

The Method of A-Harmonic Approximation and Boundary Regularity for Nonlinear Elliptic Systems under the Natural Growth Condition

We consider the questions of boundary regularity for weak solutions of second-order nonlinear elliptic systems under the natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. The proof yields directly the optimal regularity for the solution in this neighborhood. This result is new for the situation under the natural growth conditions.

Existence and Uniqueness of Positive Solutions for a Class of Semilinear Elliptic Systems

The authors prove the uniqueness and existence of positive solutions for the semilinear elliptic system which involves nonlinearities with sublinear growth conditions.

Uniform Lipschitz Estimates of Homogenization of Elliptic Systems in Divergence Form with Dini Conditions

The paper is devoted to the homogenization of elliptic systems in divergence form.We obtain uniform interior as well as boundary Lipschitz estimates in a bounded C1,γdomain when the coefficients are Dini continuous,inhomogeneous terms are divergence of Dini continuous functions and the boundary functions have Dini continuous derivatives.The results extend Avellaneda and Lin’s work[Comm.Pure Appl.Math.,40:803-847(1987)],where Holder continuity is the main assumption on smoothness of the data.

Eigenvalues of Elliptic Systems for the Mixed Problem in Perturbations of Lipschitz Domains with Nonhomogeneous Neumann Boundary Conditions

We study eigenvalues of an elliptic operator with mixed boundary conditions on very general decompositions of the boundary.We impose nonhomogeneous conditions on the part of the boundary where the Neumann term lies in a certain Sobolev or Lp space.Our work compares the behavior of and gives a relationship between the eigenvalues and eigenfunctions on the unperturbed and perturbed domains,respectively.

Structure of Coexistence States for a Class of Quasilinear Elliptic Systems

The structure of positive solutions of the p-Laplacian systems is discussed via bifurcation theory and monotone techniques.

pde2path – A Matlab Package for Continuation and Bifurcation in 2D Elliptic Systems

pde2path is a free and easy to use Matlab continuation/bifurcation package for elliptic systems of PDEs with arbitrary many components,on general two dimensional domains,and with rather general boundary conditions.The package is based on the FEM of the Matlab pdetoolbox,and is explained by a number of examples,including Bratu’s problem,the Schnakenberg model,Rayleigh-Bénard convection,and von Karman plate equations.These serve as templates to study new problems,for which the user has to provide,via Matlab function files,a description of the geometry,the boundary conditions,the coefficients of the PDE,and a rough initial guess of a solution.The basic algorithm is a one parameter arclength-continuation with optional bifurcation detection and branch-switching.Stability calculations,error control and mesh-handling,and some elementary timeintegration for the associated parabolic problem are also supported.The continuation,branch-switching,plotting etc are performed via Matlab command-line function calls guided by the AUTO style.The software can be downloaded from www.staff.uni-oldenburg.de/hannes.ue ker/pde2path,where also an online documentation of the software is provided such that in this paper we focus more on the mathematics and the example systems.

The Applications of Perturbations on Accretive Mappings to Nonlinear Elliptic Systems Related to Generalized(p,q)-Laplacian

Using perturbation results on the sums of ranges of nonlinear accretive mappings of Calvert and Gupta, we present some abstract results for the existence of the solutions of nonlinear Neumann elliptic systems which is related to the so-called generalized （p, q）-Laplacian in this paper. The systems discussed in this paper and the method used extend and complement some of the previous work.

PARTIAL REGULARITY RESULT OF SUPERQUADRATIC ELLIPTIC SYSTEMS WITH DINI CONTINUOUS COEFFICIENTS

We consider the partial regularity for weak solutions to superquadratic elliptic systems with controllable growth condition, under the assumption of Dini continuous coefficients. The proof relies upon an iteration scheme of a decay estimate for a new type of excess functional. To establish the decay estimate, we use the technique of A-harmonic approximation and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particular, the proof yields directly the optimal H¨older exponent for the derivative of the weak solutions on the regular set.