Radial Solutions For The Laplace Equation And P-Laplace Equation With A Derivative Term

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 we study the radial solutions for the Laplace equation and p-Laplace equation with a derivative term.First, we study the existence of radial positive solutions for the boundary value problem of Laplace equation When‖f‖LI’(B1)≤1, for some p> n/2, B1is a unit ball in,Rn. We achieve the supremum of u(0) by Holder inequality, and then use the spherical coordinate transformation to proof all positive solutions are radial symmetry.Second, we study the existence and uniqueness of radial solutions for the p-Laplace equation with a derivative term Δpu(t)+h(u)\u\P=f(t,u(t)),t∈(0,∞), Where/is a non-negative continuous function, u depending only ont=|x|, x∈Rn and h is a continuous function from[0,∞)to[0,∞). Using variable substitution to change the p-Laplace equation with a derivative term to the following p-Laplace equation Δpv(t)=g(t,v), t∈(0,∞),When the function g (t, v) satisfies certain conditions, we use iterative method to prove the existence of positive solutions for initial value problem of the p-Laplace equation with a derivative, and then use the Schauder fixed point theorem to prove the existence and uniqueness of non-negative solution for boundary value problem of the p-Laplace equation with a derivative term.