Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =（X1,X2,... ,Xm） be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmande...Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =（X1,X2,... ,Xm） be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmander＇s condition, then the vectorfield is finitely degenerate and the sum of square operator △X =m∑j=1 X2 j is a finitely de-generate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △x on Ω.展开更多
In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion β（u）-div（α（x, Du）＋F（u）） ∈ f in fΩ, where f ∈ L1 （Ω）. A vector field a（.,....In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion β（u）-div（α（x, Du）＋F（u）） ∈ f in fΩ, where f ∈ L1 （Ω）. A vector field a（.,.） is a Carath6odory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.展开更多
This paper studies the properties of solutions of quasilinear equations involving the p-laplacian type operator in general Carnot-Caratheodory spaces. The authors show some comparison results for solutions of the rele...This paper studies the properties of solutions of quasilinear equations involving the p-laplacian type operator in general Carnot-Caratheodory spaces. The authors show some comparison results for solutions of the relevant differential inequalities and use them to get some symmetry and monotonicity properties of solutions, in bounded or unbounded domains.展开更多
We study the deterministic homogenization of nonlinear degenerate elliptic equations with nonstandard growth. One fundamental in this topic is to extend the classical compactness results of the E-convergence method to...We study the deterministic homogenization of nonlinear degenerate elliptic equations with nonstandard growth. One fundamental in this topic is to extend the classical compactness results of the E-convergence method to the Orlicz spaces. We also show that one can homogenize nonlinear Dirichlet problems in a general way by leaning on a simple abstract hypothesis contrary to what has been done in the determinstic homogenization theory.展开更多
In this paper we obtain the H61der continuity property of the solutions for a class of degenerate Schr6dinger equation generated by the vector fields：∑i,j=1^m Xj^＊（aij（x）Xiu）＋bXu=uu=0,where X = {X1,.-. ,Xm} is...In this paper we obtain the H61der continuity property of the solutions for a class of degenerate Schr6dinger equation generated by the vector fields：∑i,j=1^m Xj^＊（aij（x）Xiu）＋bXu=uu=0,where X = {X1,.-. ,Xm} is a family of C^∞ vector fields satisfying the H6rmander condition, and the lower order terms belong to an appropriate Morrey type space.展开更多
The aim of this review is to introduce some recent results in eigenvalues problems for a class of degenerate elliptic operators with non-smooth coefficients,we present the explicit estimates of the lower bound and upp...The aim of this review is to introduce some recent results in eigenvalues problems for a class of degenerate elliptic operators with non-smooth coefficients,we present the explicit estimates of the lower bound and upper bound for its Dirichlet eigenvalues.展开更多
The present paper is concerned with the eigenvalue problem for cone degenerate p-Laplacian. First the authors introduce the corresponding weighted Sobolev spaces with important inequalities and embedding properties. T...The present paper is concerned with the eigenvalue problem for cone degenerate p-Laplacian. First the authors introduce the corresponding weighted Sobolev spaces with important inequalities and embedding properties. Then by adapting LusternikSchnirelman theory, they prove the existence of infinity many eigenvalues and eigenfunctions. Finally, the asymptotic behavior of the eigenvalues is given.展开更多
The present paper is concerned with a class of quasi-linear degenerate elliptic equations.The degenerate operator arises from analysis of manifolds with singularities.The variational methods are applied here to verify...The present paper is concerned with a class of quasi-linear degenerate elliptic equations.The degenerate operator arises from analysis of manifolds with singularities.The variational methods are applied here to verify the existence of infinitely many solutions for the problem.展开更多
In this paper,the authors study the asymptotically linear elliptic equation on manifold with conical singularities-ΔBu+λu=a(z)f(u),u≥0 in R+N,where N=n+1≥3,λ>0,z=(t,x_(1),…,x_(n)),and ΔB=(t■t)^(2)+■^(2)x_(...In this paper,the authors study the asymptotically linear elliptic equation on manifold with conical singularities-ΔBu+λu=a(z)f(u),u≥0 in R+N,where N=n+1≥3,λ>0,z=(t,x_(1),…,x_(n)),and ΔB=(t■t)^(2)+■^(2)x_(1)+…+■^(2)x_(n).Combining properties of cone-degenerate operator,the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation,we obtain a positive solution under some suitable conditions on a and f.展开更多
基金partially supported by the NSFC(11631011,11626251)
文摘Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =（X1,X2,... ,Xm） be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmander＇s condition, then the vectorfield is finitely degenerate and the sum of square operator △X =m∑j=1 X2 j is a finitely de-generate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △x on Ω.
文摘In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion β（u）-div（α（x, Du）＋F（u）） ∈ f in fΩ, where f ∈ L1 （Ω）. A vector field a（.,.） is a Carath6odory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.
基金National Natural Science Foundation of China(No.10071023)MOST and Foundation for University Key TeacherShanghai Priority Academic Discipline
文摘This paper studies the properties of solutions of quasilinear equations involving the p-laplacian type operator in general Carnot-Caratheodory spaces. The authors show some comparison results for solutions of the relevant differential inequalities and use them to get some symmetry and monotonicity properties of solutions, in bounded or unbounded domains.
文摘We study the deterministic homogenization of nonlinear degenerate elliptic equations with nonstandard growth. One fundamental in this topic is to extend the classical compactness results of the E-convergence method to the Orlicz spaces. We also show that one can homogenize nonlinear Dirichlet problems in a general way by leaning on a simple abstract hypothesis contrary to what has been done in the determinstic homogenization theory.
基金Supported by Natural Science Foundation of Zhejiang Province (No.Y60900359, Y6090383)Department of Education of Zhejiang Province (No.Z200803357)
文摘In this paper we obtain the H61der continuity property of the solutions for a class of degenerate Schr6dinger equation generated by the vector fields：∑i,j=1^m Xj^＊（aij（x）Xiu）＋bXu=uu=0,where X = {X1,.-. ,Xm} is a family of C^∞ vector fields satisfying the H6rmander condition, and the lower order terms belong to an appropriate Morrey type space.
基金supported by the National Natural Science Foundation of China(Grant Nos.11631011,11626251)the China Postdoctoral Science Foundation(Grant No.2020M672398).
文摘The aim of this review is to introduce some recent results in eigenvalues problems for a class of degenerate elliptic operators with non-smooth coefficients,we present the explicit estimates of the lower bound and upper bound for its Dirichlet eigenvalues.
基金This work was supported by the National Natural Science Foundation of China(Nos.11771218,11371282,11631011)the Fundamental Research Funds for the Central Universities。
文摘The present paper is concerned with the eigenvalue problem for cone degenerate p-Laplacian. First the authors introduce the corresponding weighted Sobolev spaces with important inequalities and embedding properties. Then by adapting LusternikSchnirelman theory, they prove the existence of infinity many eigenvalues and eigenfunctions. Finally, the asymptotic behavior of the eigenvalues is given.
基金supported by National Natural Science Foundation of China (Grant Nos. 11771218, 11371282 and 11631011)the Fundamental Research Funds for the Central Universities
文摘The present paper is concerned with a class of quasi-linear degenerate elliptic equations.The degenerate operator arises from analysis of manifolds with singularities.The variational methods are applied here to verify the existence of infinitely many solutions for the problem.
基金supported by the National Natural Science Foundation of China(Nos.11631011,11601402,12171183,11831009,12071364,11871387)。
文摘In this paper,the authors study the asymptotically linear elliptic equation on manifold with conical singularities-ΔBu+λu=a(z)f(u),u≥0 in R+N,where N=n+1≥3,λ>0,z=(t,x_(1),…,x_(n)),and ΔB=(t■t)^(2)+■^(2)x_(1)+…+■^(2)x_(n).Combining properties of cone-degenerate operator,the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation,we obtain a positive solution under some suitable conditions on a and f.
