A blowing-up branch of solutions for a mean field equation

We consider the equation -Delta u = lambda(e(u)/integral Omega(eu) - 1/\Omega\), u is an element of H-0(1)(Omega). If Omega is of class C-2, we show that this problem has a non-trivial solution u(lambda) for each gamma is an element of (8 pi,lambda*). The value lambda* depends on the domain and is bounded from below by 2j(0)(2)pi, where j(0) is the first zero of the Bessel function of the first kind of order zero (lambda* >= 2 j(0)(2) pi > 8 pi). Moreover, the family of solution u(lambda) blows- up as lambda --> 8 pi.