Minimization Techniques: Lack of Compactness
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.
KeywordsWeak Solution Sobolev Inequality Radial Function Unbounded Domain Weak Limit
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