Rendiconti del Seminario Matematico della Università di Padova


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Volume 125, 2011, pp. 1–14
DOI: 10.4171/RSMUP/125-1

Published online: 2011-06-30

Subcritical Approximation of the Sobolev Quotient and a Related Concentration Result

Giampiero Palatucci[1]

(1) Università degli Studi di Parma, Italy

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 $x_0\in \overline{\Omega}$.

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Palatucci Giampiero: Subcritical Approximation of the Sobolev Quotient and a Related Concentration Result. Rend. Sem. Mat. Univ. Padova 125 (2011), 1-14. doi: 10.4171/RSMUP/125-1