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

We consider the equation If Ω is of class C ², we show that this problem has a non-trivial solution u _λ for each λ ∊ (8 π, λ^*). The value λ^* depends on the domain and is bounded from below by 2 j ₀ ² π, where j ₀ is the first zero of the Bessel function of the first kind of order zero (λ^*≥ 2 j ₀ ² π > 8 π). Moreover, the family of solution u _λ blows-up as λ → 8 π.