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On some variational problem with limiting Sobolev exponent. (English) Zbl 0790.58013

Girardi, M. (ed.) et al., Progress in variational methods in Hamiltonian systems and elliptic equations. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 243, 42-51 (1992).
Let \(\Omega \subset \mathbb{R}^ n\), \(n \geq 3\), denote a bounded smooth domain and define \(q: 2n/(n-2)\). Then the author proves that for \(\varphi \in L^ q(\Omega)\), \(\varphi \neq 0\), a solution of \[ \int_ \Omega | \nabla u|^ 2dx \to \min \text{ in }\{u \in \overset {0} H^ 1(\Omega): \| u + \varphi\|_ q = \gamma\} \] can be constructed for any \(\gamma > 0\). His approach makes use of the best Sobolev constant and the corresponding extremal functions.
For the entire collection see [Zbl 0780.00007].

MSC:

58E30 Variational principles in infinite-dimensional spaces
49Q20 Variational problems in a geometric measure-theoretic setting
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