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.
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 an...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.展开更多
The paper is concerned with the multiplicity of solutions for some nonlinear elliptic equations involving critical Sobolev exponents and mixed boundary conditions.
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 o...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.展开更多
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 condition...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.展开更多
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 co...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.展开更多
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 i...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.展开更多
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...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.展开更多
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.
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).
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...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.展开更多
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 ellipt...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.展开更多
First, we review the authors＇ recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric soluti...First, we review the authors＇ recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition.展开更多
文摘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.
文摘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.
文摘The paper is concerned with the multiplicity of solutions for some nonlinear elliptic equations involving critical Sobolev exponents and mixed boundary conditions.
文摘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.
基金financed by the Alexander von Humboldt Foundationcontinued in March 2009 at the Mathematisches Forschungsinstitut Oberwolfach in the "Research in Pairs"program
文摘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.
基金Project supported by the Natural Science Foundation of Guangdong Province
文摘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.
文摘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.
基金supported by the National Science Foundation of China (10471047)the Natural Science Foundation of Guangdong Province (04020077)
文摘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.
文摘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.
文摘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).
文摘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.
基金This article contributes to the project"Systematic multi-scale modeling and analysis for geophysical flow"of the Collaborative Research Center TRR 181"Energy Transfers in Atmosphere and Ocean"funded by the Deutsche Forschungsgemeinschaft(DFG,German Research Foundation)under project number 274762653.
文摘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.
基金Supported by Natural Science Foundation of China (10631020, 10871061)the Grant for Ph.D Program of Ministry of Education of Chinasupported by Innovation Propject for the Development of Science and Technology (IHLB) (201098)
文摘First, we review the authors＇ recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition.
