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
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Abstract

Let Ω\Omega be a general, possibly non-smooth, bounded domain of RN\mathbb{R}^N, N3N\geq 3. Let 2 ⁣ ⁣= ⁣2N ⁣/(N2)\displaystyle 2^{*}\!\!=\!{2N}\,\!/{(N-2)} be the critical Sobolev exponent. We study the following variational problem

Sε=sup{Ωu2 ⁣εdx:Ωu2dx1,u=0 on Ω},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 ε\varepsilon goes to zero, by means of Γ+\Gamma^+-convergence techniques. We also show that sequences of maximizers uεu_\varepsilon concentrate energy at one point x0Ωx_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