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.

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.

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））.

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.

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.

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.

Superconvergence and L^(∞)-Error Estimates of the Lowest Order Mixed Methods for Distributed Optimal Control Problems Governed by Semilinear Elliptic Equations

In this paper, we investigate the superconvergence property and the L∞-errorestimates of mixed finite element methods for a semilinear elliptic control problem. Thestate and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions.We derive some superconvergence results for the control variable. Moreover, we derive L^(∞)-error estimates both for the control variable and the state variables. Finally, anumerical example is given to demonstrate the theoretical results.

A Semilinear Elliptic Equation with Double Resonance

In this paper, the existence and multiplicity of a class of double resonant semilineax elliptic equations with the Dirichlet boundary value axe studied.

Exact boundary behavior of large solutions to semilinear elliptic equations with a nonlinear gradient term

This paper is concerned with exact boundary behavior of large solutions to semilinear elliptic equations △u=b(x)f(u)(C0+|▽u|q),x∈Ω,where Ω is a bounded domain with a smooth boundary in RN,C0≥0,q E [0,2),b∈Clocα(Ω) is positive in but may be vanishing or appropriately singular on the boundary,f∈C[0,∞),f(0)=0,and f is strictly increasing on [0,∞)(or f∈C(R),f(s)> 0,■s∈R,f is strictly increasing on R).We show unified boundary behavior of such solutions to the problem under a new structure condition on f.

Asymptotic Behavior of Positive Solutions of Semilinear Elliptic Equations in R^n, Ⅱ

In this paper, we will analyze further to obtain a finer asymptotic behavior of positive solutions of semilinear elliptic equations in R^n by employing the Li＇s method of energy function.

A TRIANGULAR FINITE VOLUME ELEMENT METHOD FOR A SEMILINEAR ELLIPTIC EQUATION

In this paper we extend the idea of interpolated coefficients for a semilinear problem to the triangular finite volume element method. We first introduce triangular finite volume element method with interpolated coefficients for a boundary value problem of semilinear elliptic equation. We then derive convergence estimate in Hi-norm, L2-norm and L∞-norm, respectively. Finally an example is given to illustrate the effectiveness of the proposed method.

A Quadratic Triangular Finite Volume Element Method for a Semilinear Elliptic Equation

In this paper we extend the idea of interpolated coefficients for a semilinear problem to the quadratic triangular finite volume element method.At first we introduce quadratic triangular finite volume element method with interpolated coefficients for a boundary value problem of semilinear elliptic equation.Next we derive convergence estimate in H1-norm,L2-norm and L¥-norm,respectively.Finally an example is given to illustrate the effectiveness of the proposed method.

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.

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.

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.

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.

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.

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.