๐ฟ_{๐} embedding and nonlinear eigenvalue problems for unbounded domains
Melvyn S. Berger, Martin Schechter ยท Bulletin of the American Mathematical Society ยท 1970
Let R N denote real iV-dimensional Euclidean space.Then it is a well-known fact that the imbedding of the Sobolev spacebut is definitely not compact.Consequently the theory of critical points for general isoperimetric variational problems defined over arbitrary unbounded domains in R N has been little investigated despite its importance.Indeed the usual proofs for the existence of even one critical point for such problems requires the verification of some compactness criterion (such as Condition C of Palais-Smale).For quadratic functional, such as arise in the study of the spectrum of a linear elliptic partial differential operator L of order 2m defined on R N , many compact imbedding theorems have been obtained in recent years [l], [2], [3].These results yield, in turn, interesting facts concerning the discrete spectrum of L. In this note we extend these compactness results to insure the existence of critical points for certain isoperimetric variational problems arising in the study of eigenvalue problems for nonlinear elliptic partial differential equations.The existence of stationary states for nonlinear wave equations [4] provides a natural example for which our results are useful.