On the Global Structure of Solutions to Some Semilinear Elliptic Problems

Abstract

In this paper we discuss two problems involving positive solutions of semi linear elliptic equations which are global in the sense that the problem requires the knowledge of all possible solutions. In the first problem, we will study the semi linear elliptic equation

$$ \Delta {\text{u + }}\lambda \,{ \sinh }\,{\text{u = 0}} $$

in bounded domains D ⊂ R² . This equation has sometimes been called the elliptic Sinn-Gordon equation. Of particular interest is the study of the following boundary value problem of “nonlinear eigenvalue” type:

$$\begin{array}{*{20}{c}} {\Delta u + \lambda {\text{ sinh u }} = 0{\text{ in }}R} \\ {u = 0{\text{ on }}\partial R} \\ {u \geqslant 0{\text{ in }}R} \\ \end{array}$$

((1))

where R is a rectangle in R² . This problem arises in plasma physics and also statistical mechanics as a way of modeling point vortices. However, it arises in a surprising and central way in the construction of compact surfaces of constant man curvature. This will be explained in the following section. The basic question that we discuss is the behavior of solutions as λ tends to zero.

Research supported in part by NSF grant DMS-8501952