This paper is concerned with the existence of positive solutions of the followingDirichlet problem for p-mean curvature operator with critical exponent： -div（（1 ＋｜↓△u｜^2 ）p-2/2 ↓△u） = λup-1＋μ u=q-1,u 〉...This paper is concerned with the existence of positive solutions of the followingDirichlet problem for p-mean curvature operator with critical exponent： -div（（1 ＋｜↓△u｜^2 ）p-2/2 ↓△u） = λup-1＋μ u=q-1,u 〉 0,x x∈Ω,u=0,x∈ δΩ,where u ∈ W01,P is a bounded domain in R^N（N 〉 p 〉 1） with smooth boundary δΩ, 2≤p ≤q〈p,p=Np/N-p,λ,μ〉0. It reaches the conclusions that this problem has at least one positive solution in the different cases. It is discussed the existences of positivesolutions of the Dirichlet problem for the p-mean curvature operator with critical exponentby using Nehari-type duality property firstly. As p = 2, q = p, the result is correspond tothat of Laplace operator.展开更多
In this paper,the author proves the necessary and sufficient condition for the existence of 2-harmonically and isometrically immersed curves in a 2-dimensinonal surface N∪→IE^3.
Let M be a compact m-dimensional Riemannian manifold, let d denote, its diameter, -R(R>O) the lower bound of the Ricci curvature, and λ_1 the first eigerivalue for the Laplacian on M. Then there exists a constant ...Let M be a compact m-dimensional Riemannian manifold, let d denote, its diameter, -R(R>O) the lower bound of the Ricci curvature, and λ_1 the first eigerivalue for the Laplacian on M. Then there exists a constant C_m=max{2^(1/m-1),2^(1/2)}, Such that λ_1≥π~2/d^2·1/(2-(11)/(2π~2))+11/2π~2e^cm、展开更多
基金Supported by the National Natural Science Foundation of China（10171032） Supported by the Guangdong Provincial Natural Science Foundation of China（011606）
文摘This paper is concerned with the existence of positive solutions of the followingDirichlet problem for p-mean curvature operator with critical exponent： -div（（1 ＋｜↓△u｜^2 ）p-2/2 ↓△u） = λup-1＋μ u=q-1,u 〉 0,x x∈Ω,u=0,x∈ δΩ,where u ∈ W01,P is a bounded domain in R^N（N 〉 p 〉 1） with smooth boundary δΩ, 2≤p ≤q〈p,p=Np/N-p,λ,μ〉0. It reaches the conclusions that this problem has at least one positive solution in the different cases. It is discussed the existences of positivesolutions of the Dirichlet problem for the p-mean curvature operator with critical exponentby using Nehari-type duality property firstly. As p = 2, q = p, the result is correspond tothat of Laplace operator.
文摘In this paper,the author proves the necessary and sufficient condition for the existence of 2-harmonically and isometrically immersed curves in a 2-dimensinonal surface N∪→IE^3.
文摘Let M be a compact m-dimensional Riemannian manifold, let d denote, its diameter, -R(R>O) the lower bound of the Ricci curvature, and λ_1 the first eigerivalue for the Laplacian on M. Then there exists a constant C_m=max{2^(1/m-1),2^(1/2)}, Such that λ_1≥π~2/d^2·1/(2-(11)/(2π~2))+11/2π~2e^cm、
