Existence Of Positive Solutions For Elliptic Partial Differential Equations In The Spherical Exterior Domains

Elliptic partial differential equations have a wide range of applications in engineering and natural sciences. Many mathematical models can be described by partial differential equations. So, it's becoming very important to solve partial differential equations. In this paper the existence of positive solutions of elliptic partial differential equations are considered.Firstly, we study the positive solutions of boundary value problems for the second-order elliptic differential equations: Where the function f is continuous and f≤f1,f2,∫0+∞e2s f1(es)ds＜+∞,f2 is continuous on [0,+∞). We can transform the boundary value problem of elliptic partial differential equations for boundary value problems of second-order nonlinear ordinary differential equations. using the substitution of y(r)=y(‖x‖)=u(x), t=lnr, h(e')= y(r). Then, the existence of positive solutions is obtains by means of fixed point index theory.Next, we study the positive solutions of quasilinear elliptic differential equations in the external region of spherical. The specific equation is as follows: Where GA={x∈R":|x|>A}, n≥3, A>0. We can transform positive solutions of elliptic partial differential equations for positive solutions of second-order nonlinear ordinary differential equations using the substitution of y(r)= y(|x|)= u(x), , q(s)= sy(β(s)). Then, the existence of positive solutions is obtains by means of upper and lower solutions method, when the following conditions are satisfied.(1) 0≤f(x,t)≤a(|x|)b(t), t∈[0,∞), x∈GA; (2)∫A∞ra(r)dr<∞；(3)|g(t)|≤K|t|p,p>0,t∈R.