A logarithmic Adams-Onofri inequality and associated mean field equation

In this paper, we establish some logarithmic inequality for Sobolev spaces with second order derivatives in an arbitrary (bounded or unbounded) domain in $ \mathbb{R}^N, \ N \geq 3. $ To the very best knowledge of the author, this is the first extension of Moser-Onofri inequality known for the case of first order Sobolev space to the higher order case. Proving such a result involves many difficulties especially due to the fact that, in contrast with the case when we deal only with the first order derivatives, the symmetrization technique to reduce the general problem to the radial case cannot be used in a direct way. So, some sophisticated arguments have to employed. Taking profit of this inequality, we will be able to treat in the last part of this work some nonlinear biharmonic elliptic mean field equation of Liouville type defined in the whole space $ \mathbb{R}^N. $