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Minimization Techniques: Lack of Compactness

  • Marino Badiale
  • Enrico Serra
Part of the Universitext book series (UTX)

Abstract

In this chapter we present some examples of problems where compactness is not guaranteed a priori. The lack of compactness can take different forms, but in the simplest case, it is manifest through the fact that minimizing sequences are maybe bounded, but not (pre-)compact in the function spaces where the problem is set.

The reasons for this often come from geometrical or physical aspects, for instance when the problem is set on an unbounded domain.

Here we confine ourselves to some more or less simple examples of problems with lack of compactness and we try to show some ways to overcome the obstacle.

As usual, we study equations with different hypotheses on the nonlinearity. In particular when dealing with critical growth problems we consider homogeneous nonlinearities through minimization on spheres, while in the first sections we study nonhomogeneous nonlinearities, applying the method of minimization on the Nehari manifold.

Keywords

Weak Solution Sobolev Inequality Radial Function Unbounded Domain Weak Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di TorinoTorinoItaly

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