Functional Variable Separation for Extended Nonlinear Elliptic Equations

This paper is devoted to the study of functional variable separation for extended nonlinear elliptic equations. By applying the functional variable separation approach to extended nonlinear elliptic equations via the generalized conditional symmetry, we obtain complete classification of those equations which admit functional separable solutions （FSSs） and construct some exact FSSs to the resulting equations.

MULTIPLICITY OF SOLUTIONS AND SOURCE TERMS IN A FOURTH ORDER NONLINEAR ELLIPTIC EQUATION

The authors investigatc relations between multiplicity of solutions and sourceterms of the fourth order nonlinear elliptic boundary value problem under Dirichlet boundary condition △2u＋c△u = bu+＋f inΩ, wherc Ω is a bounded open set in Rn with smoothbonndary and the nonlinearity bu+ crosses eigenvalues of △2 ＋c△. They investigate therelatiolls when the source term is constant and when it is generated by two eigenfuntions.

MULTIPLE AND SIGN-CHANGING SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATION WITH CRITICAL POTENTIAL AND CRITICAL PARAMETER

Some embedding inequalities in Hardy-Sobolev space are proved. Furthermore, by the improved inequalities and the linking theorem, in a new k-order SobolevHardy space, we obtain the existence of sign-changing solutions for the nonlinear elliptic equation {-△（k）u：=-△u-（N-2）2/4u/｜x｜2-1/4k-1∑im1u/｜x｜2（ln（i）R/｜x｜2=f（x,u）,x∈Ω,u=0,x∈Ω,where 0∈ΩBa（0）RN,n≥3,ln）i）=6jm1ln（j）,and R=ae（k-1）,where e（0）=1,e（j）=ee（j=1）for j≥1,ln（1）=ln,ln（j）=lnln（j-1）for j≥2.Besides,positive and negative solutions are obtained by a variant mountain pass theorem.

EIGENFUNCTIONS OF THE NONLINEAR ELLIPTIC EQUATION WITH CRITICAL EXPONENT IN R^2

We consider the following eigenvalue problem: [GRAPHICS] Where f(x, t) is a continuous function with critical growth. We prove the existence of nontrivial solutions.

INFINITELY MANY SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION INVOLVING CRITICAL SOBOLEV EXPONENT

In this paper, it is proved that the following boundary value problem [GRAPHICS] admits infinitely many solution for 0 < lambda < lambda-1, n greater-than-or-equal-to 5 and for ball regions OMEGA = B(R)(0).

EXISTENCE OF MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS WITH MIXED BOUNDARY CONDITIONS

The paper is concerned with the multiplicity of solutions for some nonlinear elliptic equations involving critical Sobolev exponents and mixed boundary conditions.

UNIFORM HLDER ESTIMATES FOR A TYPE OF NONLINEAR ELLIPTIC EQUATIONS WITH RAPIDLY OSCILLATORY COEFFICIENTS

In this paper, a type of nonlinear elliptic equations with rapidly oscillatory co- efficients is investigated. By compactness methods, we show uniform HSlder estimates of solutions in a C1 bounded domain.

Solving Cauchy Issues of Highly Nonlinear Elliptic Equations Using a Meshless Method

In this paper,we address 3D inverse Cauchy issues of highly nonlinear elliptic equations in large cuboids by utilizing the new 3D homogenization functions of different orders to adapt all the specified boundary data.We also add the average classification as an approximate solution to the nonlinear operator part,without requiring to cope with nonlinear equations to resolve the weighting coefficients because these constructions are owned many conditions about the true solution.The unknown boundary conditions and the result can be retrieved straightway by coping with a small-scale linear system when the outcome is described by a new 3D homogenization function,which is right to find the numerical solutions with the errors smaller than the level of noise being put on the over-specified Neumann conditions on the bottom of the cuboid.Besides,note that the new homogenization functions method(HFM)does not require dealing with the regularization and highly nonlinear equations.The robustness and accuracy of the HFM are verified by comparing the recovered results of several numerical experiments to the exact solutions in the entire region,even though a very large level of noise 50%is imposed on the over specified Neumann conditions.The numerical errors of our scheme are in the order of O(10^(−1))-O(10^(−4)).

SOLVABILITY FOR NONLINEAR ELLIPTIC EQUATION WITH BOUNDARY PERTURBATION

The solvability of nonlinear elliptic equation with boundary perturbation is consid- ered. The perturbed solution of original problem is obtained and the uniformly valid expansion of solution is proved.

POSITIVE RADIAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS IN R^n

By the Schauder-Tychonoff fixed-point theorem, we inyestigate the existence and asymptotic behavior of positive radial solutions of fully nonlinear elliptic equations in Re. We give some sufficient conditions to guarantee the existence of bounded and unbounded radial solutions and consider the nonexistence of positive solution in Rn.

Existence of Explosive Solutions for a Class of Nonlinear Elliptic Equations

By the subsuper solutions method, the explosive supersolutions and explosive subsol utions are obtained and the exsistence of explosive solutions is proved on a bounded domain for a class of nonlinear elliptic problems.Then, the exsitence of an entire large solution is proved by the perturbed method.

AN IRREGULAR OBLIQUE DERIVATIVE PROBLEM FOR SOME NONLINEAR ELLIPTIC EQUATIONS OF SECOND ORDER

This paper deals the irregular oblique derivative problem for some nonlinear elliptic equations of second order. First a priori estimates of solutions are given, afterwards by using the above estimates of solutions and the Schauder fixed-point theorem, the existence of solutions for the above boundary value problems is proved.

SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER

The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem. Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension, the existence of solutions of the above problem is proved. In this article, the complex analytic method is used, namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed, afterwards the above problem for the degenerate elliptic equations of second order is solved.

SOME NONLINEAR ELLIPTIC EQUATIONS HAVE ONLY CONSTANT SOLUTIONS

We study some nonlinear elliptic equations on compact Riemannian manifolds. Our main concern is to find conditions which imply that such equations admit only constant solutions.

A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR,UNIFORMLY ELLIPTIC EQUATIONS

We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear, uniformly elliptic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain. In particular we de- rive sharp results on local and global Lipschitz continuity of continuous viscosity solutions under more general growth conditions than before. Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition is satisfied in a classical and not just in a viscosity sense, where detachment can occur. Another consequence is this： if interior gra- dient blow up occurs, Perron-type solutions can in general become discontinuous, so that the Dirichlet problem can become unsolvable in the class of continuous viscosity solutions.

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.

The oblique derivative problem for nonlinear elliptic complex equations of second order in multiply connected unbounded domains

In this article,we discuss that an oblique derivative boundary value problem for nonlinear uniformly elliptic complex equation of second order with the boundary conditions in a multiply connected unbounded domain D.The above boundary value problem will be called Problem P.Under certain conditions,by using the priori estimates of solutions and Leray-Schauder fixed point theorem,we can obtain some results of the solvability for the above boundary value problem（0.1） and（0.2）.

L∞-Bounds of Solutions for Strongly Nonlinear Elliptic Problems with Two Lower Order Terms

In this work, we will prove the existence of bounded solutions m W0＇ （f） N L （fl） for nonlinear elliptic equations - div（a（x,u, Vu）） ＋g（x,u,Vu） ＋ H（x, Vu） = f, where a, g and H are Carath6odory functions which satisfy some conditions, and the rizht hand side f belongs to W-l＇q （Ω）.

The Uniqueness and Nonexistent Results for Some Nonlinear Partial Equations on Riemannian Manifolds

The paper studies a class of nonlinear elliptic partial differential equations on a compact Riemannian manifold （M,g） with some curvature restriction. The authors try to prove some uniqueness and nonexistent results for the positive solutions of the equations concerned.

ON THE EXISTENCE OF SOLUTIONS TO A BI-PLANAR MONGE-AMPèRE EQUATION

In this article,we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampere operators acting in different two-dimensional coordinate sections.This equation is elliptic,for example,in the class of convex functions.We show that the notion of Monge-Ampere measures and Aleksandrov generalized solutions extends to this equation,subject to a weaker notion of convexity which we call bi-planar convexity.While the equation is also elliptic in the class of bi-planar convex functions,the contrary is not necessarily true.This is a substantial difference compared to the classical Monge-Ampere equation where ellipticity and convexity coincide.We provide explicit counter-examples:classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced.We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.