Existence and non-existence of radial solutions for a class of fourth order elliptic PDE arising in epitaxial growth theory

Abstract

In this paper, we focus on a class of fourth order elliptic partial differential equation arising in epitaxial growth theory as follows${\Delta }^{2}f=\det \left(D^{2}f\right)+\lambda G\left(x\right),x\in \Omega \subset R^2,$ where $\left(D^{2}f\right)$ is the Hessian matrix, $\lambda \in R$ is the parameter which measures the speed of the particle and $G\left(x\right)$ is the deposition rate. We fix the problem on the disk with radius T and it is defined by $\Omega =\left\{\right(x_1,x_2):{x_1}^2+{x_2}^2\le T^2\}\subset R^2$. We investigate the radial solutions subject to different types of boundary condition. Since the radial problems are nonlinear, non-self-adjoint, fourth order and a parameter λ is present, therefore it is not easy to analyze the radial solution. Here, we apply monotone iterative technique to show the existence of at least one solution in continuous space. We manifest some properties of the solutions and provide bounds for the values of the parameter λ to separate the existence from non-existence of the radial solution. Exact solution of this problem is not known. To find the approximate solutions, we develop an iterative technique based on Adomian polynomial and Green's function. We place some numerical data that will verify the theoretical results.