Existence Of Positive Solutions For Nonlinear Elliptic Equations In Spherical Domains

Elliptic partial differential equation is a branch of mathematics. It is widely applied in physics, engineering and natural science. Along with the development of theory and the progress of science and technology, elliptic partial differential equations are more extensive concerned. And the existence of positive solutions for second order elliptic partial differential equation has also got further research. In this paper, there are two parts about the main contents.1.The positive solutions of the following boundary value problems for elliptic partial equation are discussed: where k>n,Ω is a ball noted as B(o,p) with o as its center and p radius, and p is a limited real number.Firstly, by using variable substitutiony(r)=y(|x|)=u(x), we can transform the equation(2.1) for an ordinary differential equation u＂(t)+k ku’(t)/t=c(t)g(u(t)). Secondly, define a functional J whose critical point is equivalent to the solutions of the ordinary function above. Then the existence of positive solutions of (2.1)will be obtained through discussing the critical point of the functional and using P.S. conditions.2.We study the positive solutions of elliptic differential equations in spherical region. The equation is as follows: where GA={χ|χ∈R＂,|χ|≤1},n≥3.Firstly, we make the similar substitution to this equation as above. And we can also transform the equation (3.1) for ordinary differential equation y＂+g [t,u (t))=0. Then we can get two inequalities as follows:We can get a positive upper solution and a positive lower solution of equation (3.1). Then there exists at least one positive solution of (3.1) by means of upper and lower solutions method.