This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to pro...This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained.展开更多
This article considers the Dirichlet problem of homogeneous and inhomogeneous second-order ordinary differential systems. A nondegeneracy result is proven for positive solutions of homogeneous systems. Sufficient and ...This article considers the Dirichlet problem of homogeneous and inhomogeneous second-order ordinary differential systems. A nondegeneracy result is proven for positive solutions of homogeneous systems. Sufficient and necessary conditions for the existence of multiple positive solutions for inhomogeneous systems are obtained by making use of the nondegeneracy and uniqueness results of homogeneous systems.展开更多
The Allen-Cahn equation on the plane has a 6-end solution U with regular triangle symmetry. The angle between consecutive nodal lines of U is . We prove in this paper that U is non-degenerated in the class of function...The Allen-Cahn equation on the plane has a 6-end solution U with regular triangle symmetry. The angle between consecutive nodal lines of U is . We prove in this paper that U is non-degenerated in the class of functions possessing regular triangle symmetry. As an application, we show the existence of a family of solutions close to U.展开更多
We consider the boundary blow up problem for k-hessian equation with nonlinearities of power and of exponential type, and prove their existence, uniqueness and asymptotic behaviour. Moreover we also show that their pe...We consider the boundary blow up problem for k-hessian equation with nonlinearities of power and of exponential type, and prove their existence, uniqueness and asymptotic behaviour. Moreover we also show that their perturbed problem has a unique positive solution, which satisfies some asymptotic behaviors to unperturbed problems under appropriate structure hypotheses for perturbed terms.展开更多
We consider a class of analytic area-preserving mappings Cm-smoothly depending on a parameter.Without imposing on any non-degeneracy assumption,we prove a formal KAM theorem for the mappings,which implies many previou...We consider a class of analytic area-preserving mappings Cm-smoothly depending on a parameter.Without imposing on any non-degeneracy assumption,we prove a formal KAM theorem for the mappings,which implies many previous KAM-type results under some non-degeneracy conditions.Moreover,by this formal KAM theorem,we can also obtain some new interesting results under some weaker non-degeneracy conditions.Thus,the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.展开更多
In this article, nondegeneracy of singular vector tuples, Z-eigenvectors and eigenvectors of tensors is studied, which have found many applications in diverse areas. The main results are:(ⅰ)each(Z-)eigenvector/singul...In this article, nondegeneracy of singular vector tuples, Z-eigenvectors and eigenvectors of tensors is studied, which have found many applications in diverse areas. The main results are:(ⅰ)each(Z-)eigenvector/singular vector tuple of a generic tensor is nondegenerate, and(ⅱ) each nonzero Zeigenvector/singular vector tuple of an orthogonally decomposable tensor is nondegenerate.展开更多
In this paper,we consider a cone problem of matrix optimization induced by spectral norm(MOSN).By Schur complement,MOSN can be reformulated as a nonlinear semidefinite programming(NLSDP)problem.Then we discuss the con...In this paper,we consider a cone problem of matrix optimization induced by spectral norm(MOSN).By Schur complement,MOSN can be reformulated as a nonlinear semidefinite programming(NLSDP)problem.Then we discuss the constraint nondegeneracy conditions and strong second-order sufficient conditions of MOSN and its SDP reformulation,and obtain that the constraint nondegeneracy condition of MOSN is not always equivalent to that of NLSDP.However,the strong second-order sufficient conditions of these two problems are equivalent without any assumption.Finally,a sufficient condition is given to ensure the nonsingularity of the Clarke’s generalized Jacobian of the KKT system for MOSN.展开更多
基金the National Natural Science Foundation of China(Grant Nos.10671064,10171029)
文摘This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained.
基金supported by the NNSF of China(10671064)the second author was supported by the Australian Research Council's Discovery Projects(DP0450752)
文摘This article considers the Dirichlet problem of homogeneous and inhomogeneous second-order ordinary differential systems. A nondegeneracy result is proven for positive solutions of homogeneous systems. Sufficient and necessary conditions for the existence of multiple positive solutions for inhomogeneous systems are obtained by making use of the nondegeneracy and uniqueness results of homogeneous systems.
文摘The Allen-Cahn equation on the plane has a 6-end solution U with regular triangle symmetry. The angle between consecutive nodal lines of U is . We prove in this paper that U is non-degenerated in the class of functions possessing regular triangle symmetry. As an application, we show the existence of a family of solutions close to U.
基金Supported,in part,by the National Natural Science Foundation of China(No.11401060)the science and Technology Research program of Chongqing Municipal Education Commission(Grant NO.KJ 1705136)
文摘We consider the boundary blow up problem for k-hessian equation with nonlinearities of power and of exponential type, and prove their existence, uniqueness and asymptotic behaviour. Moreover we also show that their perturbed problem has a unique positive solution, which satisfies some asymptotic behaviors to unperturbed problems under appropriate structure hypotheses for perturbed terms.
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.11871146,11671077)the Innovation Project for college postgraduates in Jiangsu Province(No.KYZZ160113).
文摘We consider a class of analytic area-preserving mappings Cm-smoothly depending on a parameter.Without imposing on any non-degeneracy assumption,we prove a formal KAM theorem for the mappings,which implies many previous KAM-type results under some non-degeneracy conditions.Moreover,by this formal KAM theorem,we can also obtain some new interesting results under some weaker non-degeneracy conditions.Thus,the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.
基金supported by National Natural Science Foundation of China(Grant No.11771328)Young Elite Scientists Sponsorship Program by Tianjin and the Natural Science Foundation of Zhejiang Province of China(Grant No.LD19A010002)。
文摘In this article, nondegeneracy of singular vector tuples, Z-eigenvectors and eigenvectors of tensors is studied, which have found many applications in diverse areas. The main results are:(ⅰ)each(Z-)eigenvector/singular vector tuple of a generic tensor is nondegenerate, and(ⅱ) each nonzero Zeigenvector/singular vector tuple of an orthogonally decomposable tensor is nondegenerate.
基金This research is mainly supported by the National Natural Science Foundation of China(Nos.91130007 and 91330206).
文摘In this paper,we consider a cone problem of matrix optimization induced by spectral norm(MOSN).By Schur complement,MOSN can be reformulated as a nonlinear semidefinite programming(NLSDP)problem.Then we discuss the constraint nondegeneracy conditions and strong second-order sufficient conditions of MOSN and its SDP reformulation,and obtain that the constraint nondegeneracy condition of MOSN is not always equivalent to that of NLSDP.However,the strong second-order sufficient conditions of these two problems are equivalent without any assumption.Finally,a sufficient condition is given to ensure the nonsingularity of the Clarke’s generalized Jacobian of the KKT system for MOSN.
基金Supported by National Science Foundation of China (10771162)Foundation of Shang-hai Municipal Education Commission (071605123)Shanghai Excellent Young Teacher Foundation(SDL08015)
