A SYSTEM OF SUPERLINEAR ELLIPTIC EQUATIONS IN A CYLINDER

This dissertation is primarily concerned with the study of existence of positive solutions of nonlinear elliptic boundary value problems. An important way to deal with the problem is the study of a priori estimates of positive solutions. We will adapt a classical idea which was introduced by Brezis and Turner and, together with a fixed point theorem, we will derive the existence result of a superlinear elliptic system which defined on a cylinder. First we present some of the history of this problem, along with the necessary mathematical background. We present the main technical tools: Hardy's inequality, regularity theory and maximum principle as well as the work of Brezis and Turner. They treated a general superlinear elliptic problem and obtained the existence of positive solutions for nonlinear term having an asymptotic growth $s^\gamma$ with $1<\gamma<\frac{n+1}{n-1}$. In the novel part we apply Brezis and Turner's technique to a specific elliptic system. We study the $L^p$ regularity theory, Hardy's inequality on a cylinder and with growth conditions imposed on the nonlinear term. In particular, we will find that the nonlinear term embeds into different $L^p$ spaces as the dimension $n$ varies. We point out that there is a regularizing effect in the system which leads to a larger exponent than the Brezis-Turner exponent.