JournalsrsmupVol. 125pp. 1–14

Subcritical Approximation of the Sobolev Quotient and a Related Concentration Result

Giampiero Palatucci
Università degli Studi di Parma, Italy

Subcritical Approximation of the Sobolev Quotient and a Related Concentration Result cover

Abstract

Let $\Omega$ be a general, possibly non-smooth, bounded domain of $\mathbb{R}^N$ , $N\geq 3$ . Let $\displaystyle 2^{*}\!\!=\!{2N}\,\!/{(N-2)}$ be the critical Sobolev exponent. We study the following variational problem

S^{*}_{\varepsilon}=\sup\left \{ \int_{\Omega}|u|^{2^{*}\!-\varepsilon}dx: \int_{\Omega}|\nabla u|^{2}dx\leq 1, u=0 \ \text{on} \ \partial\Omega \right \},

investigating its asymptotic behavior as $\varepsilon$ goes to zero, by means of $\Gamma^+$ -convergence techniques.We also show that sequences of maximizers $u_\varepsilon$ concentrate energy at one point $x_0\in \overline{\Omega}$ .

Cite this article

Giampiero Palatucci, Subcritical Approximation of the Sobolev Quotient and a Related Concentration Result. Rend. Sem. Mat. Univ. Padova 125 (2011), pp. 1–14

DOI 10.4171/RSMUP/125-1