Entire solutions of semilinear elliptic equations
The present book is devoted to the study of semilinear elliptic equations of the type \[ -\Delta u+ f(x,u)= 0\quad\text{in }\mathbb{R}^N, \] where \(f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}\) has appropriate regularity. Equations of this type in bounded domains have been studied extensively. However, in unbounded domains the lack of compactness of embedding leads to new difficulties and different properties. It is the main goal of this book to describe some (existing) methods for solving the above equation in unbounded domains and illustrate the techniques by some examples. The book is organized as follows: The first chapter is devoted to classical variational methods and starts with a short introduction to the notion of generalized solutions. Some regularity results (without proof) explain the connection between generalized solutions and classical ones. The possibility to approximate the aforementioned problem by problems in bounded domains is considered as well as the monotonicity method. In Chapter 2 variational problems for eigenvalues are considered. In Chapter 3 the mountain pass method together with the concentration compactness method are applied to non-coercive elliptic problems on \(\mathbb{R}^N\). Chapter 4 presents the basic ODE techniques in the case where \(f\) is a radial function of \(x\). Moreover, a classification of the solution behavior on \(\mathbb{R}^N\) for autonomous equations is given and the phase plane method is described by means of the physically important Emden-Fowler equation. Chapter 5 yields an approach to the method of upper and lower solutions and illustrates the Leray-Schauder method. This chapter concludes with the presentation of the fibering method and some examples of nonexistence theorems. The book ends with some appendices containing results from classical functional analysis.
scientific article; zbMATH DE number 1070562