By using the Bochner type inequality of submanifold in Euclidean space,combining with the L~2-Sobolev inequality on minimal submanifold,the author generalizes Ni Lei\'s rigidity result of the minimal hypersurface with finite total scalar curvature to the case of minimal submanifold,and also gets a curvature estimate.

By using the Sobolev inequality and Gradient estimates we have proved some rigidity theorems for the space-like submanifold in a de Sitter space to be totally umbilical under the global pinching conditions of second fundamental quantity on submanifolds.

In this paper,the equivalence of the Gagliardo-Nirenberg-Sobolev inequality and the isoperimetric inequality on the Heisenberg group Hn are studied and the proof of the equivalence is given.

In this paper, we study the equivalence of the Gagliardo-Nirenberg-Sobolev inequality and the isoperimetric inequality on the Heisenberg group H", also give the proof of the equivalence.

Sobolev inequalities also called Sobolev imbedding theorems, are very popular among writers in partial differential equations or in the calculus of variations.

(Using) Sobolev-Hardy inequality,Mountain Pass lemma and Concentration compactness principle,the existence of positive solution was proved under the certain conditions that the cofficients and exponents of the(equation) meet.

The authors discuss the existence of positive solution for a p-Laplace equation with singular weight by using Sobolev-Hardy inequality and the Mountain Pass Lemma.

We use the decomposition of the filtration of the Nehair manifold via the variation of domain shape and Sobolev-Hardy inequality.