Problems with Lack of Compactness
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 ,
is invariant under the group of translations u(·) ↦ u (· + z), z ∈ ℝ N . We will be considering one such situation in this chapter.
$$ - \Delta u + u = h\left( u \right), $$
KeywordsSobolev Inequality Translation Invariance Strict Convexity Pure Power Unique Critical Point
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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© Birkhäuser Boston 2007