ELLIPTIC EQUATIONS IN DIVERGENCE FORM WITH DISCONTINUOUS COEFFICIENTS IN DOMAINS WITH CORNERS

We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C1,αestimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners.

The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.

Besov Estimates for Sub-Elliptic Equations in the Heisenberg Group

In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.

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.

GLOBAL BOUND ON THE GRADIENT OF SOLUTIONS TO p-LAPLACE TYPE EQUATIONS WITH MIXED DATA

In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.

SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS

Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem△_(p)u=b(x)g(u)for x∈Ω,u(x)→+∞as dist(x,■Ω)→0.The estimates of such solutions are also investigated.Moreover,when b has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.

Resonant interactions among two-dimensional nonlinear localized waves and lump molecules for the(2+1)-dimensional elliptic Toda equation

The(2+1)-dimensional elliptic Toda equation is a high-dimensional generalization of the Toda lattice and a semidiscrete Kadomtsev–Petviashvili I equation.This paper focuses on investigating the resonant interactions between two breathers,a breather/lump and line solitons as well as lump molecules for the(2+1)-dimensional elliptic Toda equation.Based on the N-soliton solution,we obtain the hybrid solutions consisting of line solitons,breathers and lumps.Through the asymptotic analysis of these hybrid solutions,we derive the phase shifts of the breather,lump and line solitons before and after the interaction between a breather/lump and line solitons.By making the phase shifts infinite,we obtain the resonant solution of two breathers and the resonant solutions of a breather/lump and line solitons.Through the asymptotic analysis of these resonant solutions,we demonstrate that the resonant interactions exhibit the fusion,fission,time-localized breather and rogue lump phenomena.Utilizing the velocity resonance method,we obtain lump–soliton,lump–breather,lump–soliton–breather and lump–breather–breather molecules.The above works have not been reported in the(2+1)-dimensional discrete nonlinear wave equations.

The Jacobi elliptic function-like exact solutions to two kinds of KdV equations with variable coefficients and KdV equation with forcible term

By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg de Vries （KdV） equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.

INTERIOR ESTIMATES IN MORREY SPACES FOR SOLUTIONS OF ELLIPTIC EQUATIONS AND WEIGHTED BOUNDEDNESS FOR COMMUTATORS OF SINGULAR INTEGRAL OPERATORS

It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp, (w).

Solving Nonlinear Wave Equations by Elliptic Equation

The elliptic equation is taken as a transformation and applied to solve nonlinear wave equations. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions,periodic wave solutions and so on, so it can be taken as a generalized method.

Construction of doubly-periodic solutions to nonlinear partial differential equations using improved Jacobi elliptic function expansion method and symbolic computation

Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly-periodic solutions of the Zakharov-Kuznetsov equation, which has been derived by Gottwald as a two-dimensional model for nonlinear Rossby waves. When the modulus k →1, these solutions reduce to the solitary wave solutions of the equation.

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.

THE SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER SEMILINEAR ELLIPTIC EQUATIONS

In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansion in entire region is obtained.

The Nonlinear Nonlocal Singularly Perturbed Problems for Elliptic Equations with Boundary Perturbation

The nonlinear nonlocal singularly perturbed boundary value problems for elliptic equation with boundary perturbation was considered.Under suitable conditions,firstly,the outer solution of the original problem is obtained,secondly,using the stretched variable,the composing expansion method and the expanding theory of power series the boundary layer is constructed,finally,using the theory of differential inequalities the asymptotic behavior of solution for the boundary value problems is studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed.

Applications of Jacobi Elliptic Function Expansion Method for Nonlinear Differential-Difference Equations

The Jacobi elliptic function expansion method is extended to derive the explicit periodic wave solutions for nonlinear differential-difference equations. Three well-known examples are chosen to illustrate the application of the Jacobi elliptic function expansion method. As a result, three types of periodic wave solutions including Jacobi elliptic sine function, Jacobi elliptic cosine function and the third elliptic function solutions are obtained. It is shown that the shock wave solutions and solitary wave solutions can be obtained at their limit condition.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF SECOND ORDER NONLINEAR ELLIPTIC DIFFERENTIAL EQUATIONS

In this paper, the second order nonlinear elliptic differential equations (E) (n)Sigma (i,j=1) partial derivative/partial derivativex(j)[a(i,j)(x,y) partial derivative/partial derivativex(j)y] + q(x)f(y) = e(x) are considered in an exterior Omega subset of R-n, where q(x) is allowed to change sign. Some sufficient conditions for any solutions y(x) of (E) to be satisfied liminf\\x\--> infinity \y(x)\ = 0 are obtained. Particularly, these results improve the previous results for second order ordinary differential equations.

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.

Convergence of gradient of solutions to elliptic equations with nonuniform growth

In this paper, we study elliptic partial differential equations with nonuniform growth, and obtain the strong convergence of the gradient of truncations of solutions and the strong convergence of the gradient of solutions in the setting of some Musielak-Orlicz spaces. Our results are generalization of the corresponding results for elliptic partial differential equations with power growth.

MULTIPLICITY RESULTS FOR FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE

In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti- Rabinowitz type condition.

THE FIRST BOUNDARY VALUE PROBLEM FOR A CLASS OF QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS

In this paper, the first boundary value problem for quasilinear equation of the form △A（u,x）＋∑i=1^m δb^i（u,x）/δxi＋c（u,x）=0,Au（u,x） ≥0is studied. By using the compensated compactness theory, some results on the existence of weak solution are established. In addition, under certain condition the uniqueness of solution is proved.