Problems with Lack of Compactness

  • David G. Costa


In this chapter and the next we will present two examples of situations in which the variational problem under consideration lacks some desirable compactness properties. Typically, lack of compactness is due to the action of a group under which the pertinent functional is invariant. For example, an autonomous semilinear elliptic equation in the whole space ℝ N ,
$$ - \Delta u + u = h\left( u \right), $$
is invariant under the group of translations u(·) ↦ u (· + z), z ∈ ℝ N . We will be considering one such situation in this chapter.


Sobolev Inequality Translation Invariance Strict Convexity Pure Power Unique Critical Point 
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Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • David G. Costa
    • 1
  1. 1.Department of Mathematical SciencesUniversity of Nevada, Las VegasLas VegasUSA

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