Asymptotic behavior of positive solutions of quasilinear elliptic equations with critical Sobolev growth

Abstract

In this short note we investigate the asymptotic behavior of positive minimizing solutions $u_\epsilon$ to the equation $\Delta u_\epsilon = N(N-2) f(x) u_\epsilon^{p-\epsilon}$ in $\Omega$ and $u_\epsilon = 0$ on $\partial\Omega$, where $\Delta$ stands for the Euclidean Laplacian with the minus sign convention, $\Omega$ is a smooth bounded domain in ${\mathbb R}^N$, $p = (N+2)/(N-2)$ is the critical Sobolev exponent, and $f$ belongs to a fairly general class of functions.