EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO A CLASS OF SEMILINEAR ELLIPTIC SYSTEMS

In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the application of the main theorem, two examples are given.

THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R N WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

In this paper, we prove the existence of at least one positive solution pair （u, v）∈ H1（RN） × H1（RN） to the following semilinear elliptic system {-△u＋u=f（x,v）,x∈RN,-△u＋u=g（x,v）,x∈RN （0.1）,by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈C0（RN× R1） are that, f（x, t） and g（x, t） are superlinear at t = 0 as well as at t =＋∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245（2008）, 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem {-△u＋u=f（x,u）,x∈Ω,u∈H0^1（Ω） where Ω ∩→RN is bounded and a result of Li and Yang [G. Li and J. Yang： Communications in P.D.E. Vol. 29（2004） Nos.5＆ 6.pp.925-954, 2004] concerning （0.1） when f and g are asymptotically linear.

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.

A Posteriori Error Estimate of Two Grid Mixed Finite Element Methods for Semilinear Elliptic Equations

In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.

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.

AN EXISTENCE THEOREM OF SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS ON R^N

An existent theorem is obtained for nonzero W-1,W-2(R-N) solutions of the following equations on R-N -Delta u + b(x)u = f(x,u), x is an element of R-N, where b is periodic for some variables and coercive for the others, f is superlinear.

PERTURBATION METHODS IN SEMILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY-SOBOLEV EXPONENT

In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory.

SEPARATION PROPERTY OF POSITIVE RADIAL SOLUTIONS FOR A GENERAL SEMILINEAR ELLIPTIC EQUATION

The asymptotic behavior at infinity and an estimate of positive radial solutions of the equation △u ＋ sum from i=1 to k cirli upi = 0, x ∈ Rn,（0.1）are obtained and the structure of separation property of positive radial solutions of Eq. （0.1） with different initial data α is discussed.

SINGULAR POSITIVE RADIAL SOLUTIONS FOR A GENERAL SEMILINEAR ELLIPTIC EQUATION

The existence and uniqueness of singular solutions decaying like r^-m（see （1.4）） of the equation △u＋k∑i=1ci｜x｜liupi=0,x∈R^N are obtained, wheren≥3, ci 〉0, li〉-2, i=1,2,..,k, pi〉 1, i=l,2,-..,kandthe separation structure of singular solutions decaying like r^-（n-2） of eq. （0.1） are discussed. moreover, we obtain the explicit critical exponent ps （l） （see （1.9））.

Kamenev-type Oscillation Criteria for Semilinear Elliptic Differential Equations

Oscillation criteria for semilinear elliptic differential equations are obtained. The results are extensions of integral averaging technique of Kamenev. General means are employed to establish our results.

THE EXISTENCE AND GLOBAL OPTIMAL ASYMPTOTIC BEHAVIOUR OF LARGE SOLUTIONS FOR A SEMILINEAR ELLIPTIC PROBLEM

By Karamata regular variation theory and constructing comparison functions, the author shows the existence and global optimal asymptotic behaviour of solutions for a semilinear elliptic problem Δu = k（x）g（u）, u 〉 0, x ∈ Ω, u｜δΩ =＋∞, where Ω is a bounded domain with smooth boundary in R^N; g ∈ C^1[0, ∞）, g（0） = g＇（0） = 0, and there exists p 〉 1, such that lim g（sξ）/g（s）=ξ^p, ↓Aξ 〉 0, and k ∈ Cloc^α（Ω） is non-negative non-trivial in D which may be singular on the boundary.

EXISTENCE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS AND HARDY TERMS

This paper deals with the existence of solutions to the elliptic equation-△u-μ/|x|2=λu +|u|2*-2u + f(x,u) in Ω,u = 0 on (?)Ω, where Ω is a bounded domain in RN(N≥3), 0 ∈ Ω 2*=2N/N-2,λ> 0, λ (?) σμ,σμ is the spectrum of the operator -△-μI/|x|2 with zero Dirichlet boundary condition, 0 <μ< μ-,μ-=(N-2)2/4, f(x,u)is an asymmetric lower order perturbation of |u|2* -1 at infinity. Using the dual variational methods, the existence of nontrivial solutions is proved.

Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second‑Order Elliptic Problems on Cartesian Grids

This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order elliptic problems of the form−Δu=f(x,y,u)on Cartesian grids.By introducing special GaussRadau projections and using duality arguments,we obtain,under some suitable choice of numerical fuxes,the optimal convergence order in L2-norm of O(h^(p+1))for the LDG solution and its gradient,when tensor product polynomials of degree at most p and grid size h are used.Moreover,we prove that the LDG solutions are superconvergent with an order p+2 toward particular Gauss-Radau projections of the exact solutions.Finally,we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves(p+1)-th order superconvergence.Some numerical experiments are performed to illustrate the theoretical results.

ON NEUMANN PROBLEM FOR SOMESEMILINEAR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS

This paper is concerned with Neumann problem for semilinear elliptic equations involving Sobolev critical exponents with limit nonlinearity in boundary condition. By critical point theory and dual variational principle, the author obtains the existence and multiplicity results.

NEUMANN PROBLEMS OF A CLASS OF ELLIPTIC EQUATIONS WITH DOUBLY CRITICAL SOBOLEV EXPONENTS

This paper deals with the Neumann problem for a class of semilinear elliptic equations -△u + u =|u|2*-2u+ μ|u|q-2u in Ω, au/ar= |u|(?)*-2u on aΩ, where 2 = 2N/N-2, s=2(N-1)/N-2, 1 <q<2,N(?)3,μ>γ denotes the unit outward normal to boundary aΩ. By vaxiational method and dual fountain theorem, the existence of infinitely many solutions with negative energy is proved.

INFINITELY MANY SOLUTIONS FOR AN ELLIPTIC PROBLEM INVOLVING CRITICAL NONLINEARITY

We study the following elliptic problem：{-div（a（x）Du）=Q（x）｜u｜2-2u＋λu x∈Ω,u=0 onδΩ Under certain assumptions on a and Q, we obtain existence of infinitely many solutions by variational method.

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF CONCAVE-CONVEX ELLIPTIC EQUATIONS WITH CRITICAL GROWTH

In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered,{-△u=g（x）｜u｜2＊-2u＋λf（x）｜u｜q-2u,x∈Ω u=0,x∈δΩ where Ω RN（N ≥ 3） is an open bounded domain with smooth boundary, 1 〈 q 〈 2, λ 〉 0. 2＊= 2N/N-2 is the critical Sobolev exponent, f ∈L2＊/2N/N-2 is nonzero and nonnegative, and g E （Ω） is a positive function with k local maximum points. By the Nehari method and variational method, k ＋ 1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75（2012） 2660-26711.

Boundary Lipschitz Regularity of Solutions for Semilinear Elliptic Equations in Divergence Form

In this paper,we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary.If the domain satisfies C1,Dinicondition at a boundary point,and the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition,then the solution is Lipschitz continuous at this point.Furthermore,we generalize this result to Reifenberg C1,Dinidomains.

Discontinuous Galerkin Methods for Semilinear Elliptic Boundary Value Problem

A discontinuous Galerkin(DG)scheme for solving semilinear elliptic problem is developed and analyzed in this paper.The DG finite element discretization is first established,then the corresponding well-posedness is provided by using Brouwer’s fixed point method.Some optimal priori error estimates under both DG norm and L^(2)norm are presented,respectively.Numerical results are given to illustrate the efficiency of the proposed approach.

Nonexistence of positive supersolutions to a class of semilinear elliptic equations and systems in an exterior domain

In this paper,we consider the following semilinear elliptic equation:■whereΩis an exterior domain in R^N with N≥3,h:Ω×R^+→R is a measurable function,and derive optimal nonexistence results of positive supersolutions.Our argument is based on a nonexistence result of positive supersolutions of a linear elliptic problem with Hardy potential.We also establish sharp nonexistence results of positive supersolutions to an elliptic system.