Bubbling phenomena for fourth-order four-dimensional PDEs with exponential growth

We are concerned in this paper with the bubbling phenomenon for nonlinear fourth-order four-dimensional PDE's. The operators in the equations are perturbations of the bi-Laplacian. The nonlinearity is of exponential growth. Such equations arise naturally in statistical physics and geometry. As a consequence of our theorem we get a priori bounds for solutions of our equations.