Existence Of Positive Solutions For Elliptic Partial Differential Equations Involving Hardy-Sobolev Critical Exponent

The application in engineering sciences and natural sciences of partial differential equations is very extensive. Many technical problems must to be transformed into the problem solving partial differential equations. Therefore, the study of partial differential equations has important practical significance. Which has the elliptic partial differential equations involving Hardy-Sobolev critical exponent has caused widespread concern in many Mathematics workers at home and abroad, and having achieved many important results. On this basis, the article continued to study with Existence of positive solutions for elliptic partial differential equations involving Hardy-Sobolev critical exponent, divided into two parts:1. We discussed the existence of a Hardy-Sobolev critical singular elliptic equation with mixed Dirichlet-Neumann boundardy conditions. In the condition of1<q<2*, firstly, we put into the existence of equation to the existence of critical points of corresponding functional. Secondly, using Ekeland Variational principle, we got a minimum of functional sequence, and thus overcome the loss of compact embedded. Finally using a deformation of the mountain pass lemma, we proved the existence of equations.2. We discussed with the Hardy-Sobolev critical exponents of the semi-linear elliptic equations Existence of multiple positive solutions. We first put into the existence of equation to the existence of critical points of corresponding functional. Using Ekeland variational principle and the Mountain Pass Lemma, we got the first solution of equation. Then using a functional which have changed and Mountain Pass Lemma, we got the second solution equations.