Some Qualitative Properties of Solutions of Semilinear Elliptic Equations in Cylindrical Domains

Abstract

This chapter explains some qualitative properties of semilinear elliptic equations in cylindrical domains. In studying differential equations, it is often of interest to know if the solutions have symmetry, or monotonicity, in some direction. Using results of boundary conditions, monotonicity, symmetry properties, and uniqueness for some solutions in an infinite cylinder are established. When using the method of moving planes in all of space to prove monotonicity and symmetry, the asymptotic behavior of the solution near infinity is used to start the procedure. The behavior at infinity of solutions of semi-linear elliptic equations in infinite cylindrical domains is governed by some exponential solutions.