This paper deals with the existence and multiplicity of nontrivial solutions to a weighted nonlinear elliptic system with nonlinear homogeneous boundary condition in a bounded domain. By using the Caffarelli-Kohn-Nire...This paper deals with the existence and multiplicity of nontrivial solutions to a weighted nonlinear elliptic system with nonlinear homogeneous boundary condition in a bounded domain. By using the Caffarelli-Kohn-Nirenberg inequality and variational method, we prove that the system has at least two nontrivial solutions when the parameter λ belongs to a certain subset of R.展开更多
In this paper, by using the idea of category, we investigate how the shape of the graph of h（x） affects the number of positive solutions to the following weighted nonlinear elliptic system： = （ N-2-2a 2. where 0 ...In this paper, by using the idea of category, we investigate how the shape of the graph of h（x） affects the number of positive solutions to the following weighted nonlinear elliptic system： = （ N-2-2a 2. where 0 is a smooth bounded domain in ]1N （N 〉 3）, A, cr 〉 0 are parameters, 0 ≤ μ 〈 μa a 2 ＇ h（x）, KI（X） and K2（x） are positive continuous functions in , 1 〈 q 〈 2, a, β 〉 1 and a ＋ β = 2＊（a,b） （2＊ （a, b） 2N = N-2（1＋a-b） is critical Sobolev-Hardy exponent）. We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters （）, r） belongs to a certain subset of N2.展开更多
Let Ω be a bounded domain with a smooth C2 boundary in RN（N ≥ 3）, 0 ∈Ω, and n denote the unit outward normal to ЭΩ.We are concerned with the Neumann boundary problems： -div（｜x｜α｜△u｜p-2△u）=｜x｜βup（...Let Ω be a bounded domain with a smooth C2 boundary in RN（N ≥ 3）, 0 ∈Ω, and n denote the unit outward normal to ЭΩ.We are concerned with the Neumann boundary problems： -div（｜x｜α｜△u｜p-2△u）=｜x｜βup（α,β）-1-λ｜x｜γup-1,u（x）〉0,x∈Ω,Эu/Эn=0 on ЭΩ,where 1〈p〈N and α〈0,β〈0 such that p（α,β）△=p（N＋β）/N-p＋α〉p,y〉α-p.For various parameters α,βorγ,we establish certain existence results of the solutions in the case 0∈Ω or 0∈ЭΩ.展开更多
In this paper, we establish the existence of at least four distinct solutions to an elliptic problem with singular cylindrical potential, a concave term, and critical Caffarelli-Kohn-Nirenberg exponent, by using the N...In this paper, we establish the existence of at least four distinct solutions to an elliptic problem with singular cylindrical potential, a concave term, and critical Caffarelli-Kohn-Nirenberg exponent, by using the Nehari manifold and mountain pass theorem.展开更多
We prove that the limits of the semi-discrete and the discrete semi-implicit Euler schemes for the 3D Navier-Stokes equations supplemented with Dirichlet boundary conditions are suitable in the sense of Scheffer [1]. ...We prove that the limits of the semi-discrete and the discrete semi-implicit Euler schemes for the 3D Navier-Stokes equations supplemented with Dirichlet boundary conditions are suitable in the sense of Scheffer [1]. This provides a new proof of the existence of suitable weak solutions, first established by Caffarelli, Kohn and Nirenberg [2]. Our results are similar to the main result in [3]. We also present some additional remarks and open questions on suitable solutions.展开更多
In this paper a class of p-Laplace type elliptic equations with unbounded coefficients on RN is considered. It is proved that there exist radial solutions on RN. On sufficiently large ball, radial and nonradial soluti...In this paper a class of p-Laplace type elliptic equations with unbounded coefficients on RN is considered. It is proved that there exist radial solutions on RN. On sufficiently large ball, radial and nonradial solutions are obtained. Finally, some necessary conditions for the existence of solutions are given.展开更多
We observe,utilize dualities in differential equations and differential inequalities(see Theorem 2.1),dualities between comparison theorems in differential equations(see Theorems E and 2.2),and obtain dualities in&quo...We observe,utilize dualities in differential equations and differential inequalities(see Theorem 2.1),dualities between comparison theorems in differential equations(see Theorems E and 2.2),and obtain dualities in"swapping"comparison theorems in differential equations.These dualities generate comparison theorems on differential equations of mixed typesⅠandⅡ(see Theorems 2.3 and 2.4)and lead to comparison theorems in Riemannian geometry(see Theorems 2.5 and 2.8)with analytic,geometric,PDE's and physical applications.In particular,we prove Hessian comparison theorems(see Theorems 3.1-3.5)and Laplacian comparison theorems(see Theorems 2.6,2.7 and 3.1-3.5)under varied radial Ricci curvature,radial curvature,Ricci curvature and sectional curvature assumptions,generalizing and extending the work of Han-Li-Ren-Wei(2014)and Wei(2016).We also extend the notion of function or differential form growth to bundle-valued differential form growth of various types and discuss their interrelationship(see Theorem 5.4).These provide tools in extending the notion,integrability and decomposition of generalized harmonic forms to those of bundle-valued generalized harmonic forms,introducing Condition W for bundle-valued differential forms,and proving the duality theorem and the unity theorem,generalizing the work of Andreotti and Vesentini(1965)and Wei(2020).We then apply Hessian and Laplacian comparison theorems to obtain comparison theorems in mean curvature,generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds,the embedding theorem for weighted Sobolev spaces of functions on manifolds,geometric differential-integral inequalities,generalized sharp Hardy type inequalities on Riemannian manifolds,monotonicity formulas and vanishing theorems for differential forms of degree k with values in vector bundles,such as F-Yang-Mills fields(when F is the identity map,they are Yang-Mills fields),generalized Yang-Mills-Born-Infeld fields on manifolds,Liouville type theorems for Fharmonic maps(when F(t)=1/p(2 t)展开更多
文摘This paper deals with the existence and multiplicity of nontrivial solutions to a weighted nonlinear elliptic system with nonlinear homogeneous boundary condition in a bounded domain. By using the Caffarelli-Kohn-Nirenberg inequality and variational method, we prove that the system has at least two nontrivial solutions when the parameter λ belongs to a certain subset of R.
文摘In this paper, by using the idea of category, we investigate how the shape of the graph of h（x） affects the number of positive solutions to the following weighted nonlinear elliptic system： = （ N-2-2a 2. where 0 is a smooth bounded domain in ]1N （N 〉 3）, A, cr 〉 0 are parameters, 0 ≤ μ 〈 μa a 2 ＇ h（x）, KI（X） and K2（x） are positive continuous functions in , 1 〈 q 〈 2, a, β 〉 1 and a ＋ β = 2＊（a,b） （2＊ （a, b） 2N = N-2（1＋a-b） is critical Sobolev-Hardy exponent）. We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters （）, r） belongs to a certain subset of N2.
基金Supported by the National Natural Science Foundation of China(No.10631030)the Program for New Century Excellent Talents in University(No.07-0350)the Key Project of Chinese Ministry of Education(No.107081)
文摘Let Ω be a bounded domain with a smooth C2 boundary in RN（N ≥ 3）, 0 ∈Ω, and n denote the unit outward normal to ЭΩ.We are concerned with the Neumann boundary problems： -div（｜x｜α｜△u｜p-2△u）=｜x｜βup（α,β）-1-λ｜x｜γup-1,u（x）〉0,x∈Ω,Эu/Эn=0 on ЭΩ,where 1〈p〈N and α〈0,β〈0 such that p（α,β）△=p（N＋β）/N-p＋α〉p,y〉α-p.For various parameters α,βorγ,we establish certain existence results of the solutions in the case 0∈Ω or 0∈ЭΩ.
文摘In this paper, we establish the existence of at least four distinct solutions to an elliptic problem with singular cylindrical potential, a concave term, and critical Caffarelli-Kohn-Nirenberg exponent, by using the Nehari manifold and mountain pass theorem.
文摘We prove that the limits of the semi-discrete and the discrete semi-implicit Euler schemes for the 3D Navier-Stokes equations supplemented with Dirichlet boundary conditions are suitable in the sense of Scheffer [1]. This provides a new proof of the existence of suitable weak solutions, first established by Caffarelli, Kohn and Nirenberg [2]. Our results are similar to the main result in [3]. We also present some additional remarks and open questions on suitable solutions.
文摘In this paper a class of p-Laplace type elliptic equations with unbounded coefficients on RN is considered. It is proved that there exist radial solutions on RN. On sufficiently large ball, radial and nonradial solutions are obtained. Finally, some necessary conditions for the existence of solutions are given.
基金supported by National Science Foundation of USA(Grant No.DMS1447008)。
文摘We observe,utilize dualities in differential equations and differential inequalities(see Theorem 2.1),dualities between comparison theorems in differential equations(see Theorems E and 2.2),and obtain dualities in"swapping"comparison theorems in differential equations.These dualities generate comparison theorems on differential equations of mixed typesⅠandⅡ(see Theorems 2.3 and 2.4)and lead to comparison theorems in Riemannian geometry(see Theorems 2.5 and 2.8)with analytic,geometric,PDE's and physical applications.In particular,we prove Hessian comparison theorems(see Theorems 3.1-3.5)and Laplacian comparison theorems(see Theorems 2.6,2.7 and 3.1-3.5)under varied radial Ricci curvature,radial curvature,Ricci curvature and sectional curvature assumptions,generalizing and extending the work of Han-Li-Ren-Wei(2014)and Wei(2016).We also extend the notion of function or differential form growth to bundle-valued differential form growth of various types and discuss their interrelationship(see Theorem 5.4).These provide tools in extending the notion,integrability and decomposition of generalized harmonic forms to those of bundle-valued generalized harmonic forms,introducing Condition W for bundle-valued differential forms,and proving the duality theorem and the unity theorem,generalizing the work of Andreotti and Vesentini(1965)and Wei(2020).We then apply Hessian and Laplacian comparison theorems to obtain comparison theorems in mean curvature,generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds,the embedding theorem for weighted Sobolev spaces of functions on manifolds,geometric differential-integral inequalities,generalized sharp Hardy type inequalities on Riemannian manifolds,monotonicity formulas and vanishing theorems for differential forms of degree k with values in vector bundles,such as F-Yang-Mills fields(when F is the identity map,they are Yang-Mills fields),generalized Yang-Mills-Born-Infeld fields on manifolds,Liouville type theorems for Fharmonic maps(when F(t)=1/p(2 t)
