NONDEGENERACY OF POSITIVE SOLUTIONS TO HOMOGENEOUS SECOND-ORDER DIFFERENTIAL SYSTEMS AND ITS APPLICATIONS

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.

Nondegeneracy of Solution to the Allen-Cahn Equation with Regular Triangle Symmetry

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.

Nondegeneracy of eigenvectors and singular vector tuples of tensors

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.

Existence and uniqueness of positive solutions of semilinear elliptic equations

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.

Existence of invariant curves for area-preserving mappings under weaker non-degeneracy conditions

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.

Asymptotic Behaviour of Hessian Equation with Boundary Blow Up

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.

On Some Aspects of Perturbation Analysis for Matrix Cone Optimization Induced by Spectral Norm

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.