Displaying similar documents to “Subcritical approximation of the Sobolev quotient and a related concentration result”

Γ -convergence of concentration problems

Micol Amar, Adriana Garroni (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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In this paper, we use Γ -convergence techniques to study the following variational problem S ε F ( Ω ) : = sup ε - 2 * ∫ Ω F ( u ) d x : ∫ Ω | ∇ u | 2 d x ≤ ε 2 , u = 0 on ∂ Ω , where 0 ≤ F ( t ) ≤ | t | 2 * , with 2 * = 2 n n - 2 , and Ω is a bounded domain of ℝ n , n ≥ 3 . We obtain a Γ -convergence result, on which one can easily read the usual concentration phenomena arising in critical growth problems. We extend the result to a non-homogeneous version of problem S ε F ( Ω ) . Finally, a second order expansion in Γ -convergence permits to identify the concentration points of the maximizing sequences,...

On the Neumann problem for an elliptic system of equations involving the critical Sobolev exponent

J. Chabrowski, Jianfu Yang (2001)

Colloquium Mathematicae

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We consider the Neumann problem for an elliptic system of two equations involving the critical Sobolev nonlinearity. Our main objective is to study the effect of the coefficient of the critical Sobolev nonlinearity on the existence and nonexistence of least energy solutions. As a by-product we obtain a new weighted Sobolev inequality.