Asymptotic behavior of a nonlinear fourth order eigenvalue problem

Our purpose in this paper is to study the asymptotic behavior of the nonlinear eigenvalue problem, where Ω is a smooth bounded domain in R4, line-integral(u) is an nonnegative smooth function with exponentially dominant nonlinearity and λ > 0 is small. When line-integral(u) = eu, equation arises in the study of the extremal metrics of zeta function determinants on 4-manifolds. It becomes very important to study a priori estimates of solutions. Two extreme cases are λ → 0 and λ → ∞. It is easy to see that when λ is large, there is no solution to the problem. We shall study the case when λ → 0