Subcritical Approximation of the Sobolev Quotient and a Related Concentration Result

Abstract

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

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