Electronic Journal of Differential Equations,
Monograph 06, 2004 (142 pages)

Title: Palais-Smale approaches to semilinear elliptic equations
       in unbounded domains

Author: Hwai-chiuan Wang (National Tsing Hua Univ., Taiwan)

Abstract: 
  Let $\Omega$ be a domain in $\mathbb{R}^{N}$, $N\geq1$, 
  and $2^{\ast}=\infty$ if $N=1,2$, $2^{\ast}=\frac{2N}{N-2}$ 
  if  $N>2$, $2<p<2^{\ast}$. 
  Consider the semilinear elliptic  problem
  $$\displaylines{
  -\Delta u+u=|u|^{p-2}u\quad\hbox{in }\Omega;\cr
  u\in H_{0}^{1}(\Omega).
  }$$
  Let $H_{0}^{1}(\Omega)$ be the Sobolev space in $\Omega$.
  The existence, the nonexistence, and the multiplicity of positive
  solutions  are affected by the geometry and the topology
  of the domain $\Omega$. The existence, the nonexistence, and the
  multiplicity of positive solutions  have been the focus of a great
  deal of research in recent years.

  That the above equation in a bounded domain admits a positive
  solution  is a classical result. Therefore the only interesting
  domains in which this equation admits a positive solution are proper
  unbounded domains. Such elliptic problems are difficult because of
  the lack of compactness in unbounded domains. Remarkable progress
  in the study of this kind of problem has been made by P. L. Lions.
  He developed the concentration-compactness principles for solving
  a large class of minimization problems with constraints in unbounded
  domains. The characterization of domains in which this equation
  admits a positive solution is an important open question.
  In this monograph, we present various analyses and use them to
  characterize several categories of domains in which this equation
  admits a positive solution or multiple solutions.
 
Submitted September 17, 2004. Published September 30, 2004.
Math Subject Classifications: 35J20, 35J25
Key Words: Palais-Smale condition; index; decomposition theorem;
           achieved domain; Esteban-Lions domain; 
	   symmtric Palais-Smale condition.