Global compactness properties of semilinear elliptic equations with critical exponential growth (Q1580802)

It is considered the semilinear elliptic boundary value problem \(-\Delta u = f(u)\), \(u>0\) in \(\Omega\), \(u=0\) on \(\partial\Omega\). Here \(f\) is smooth and has critical exponential growth, \(\Omega\subset\mathbb R^2\). In this problem either the Palais-Smale condition holds or blow-up occur. The authors investigated both situations in \(\mathbb R^2\) and showed the difference for higher-dimensional problems. In some blow up cases they obtained a universal description of some concentration behavior.