Existence and multiplicity results for a fourth order mean field equation

Abstract

In this article we consider the following fourth order mean field equation on smooth domain Ω R 4 :where ∈ R and 0 < K ∈ C 2 (Ω). Through a refined blow up analysis, we characterize the critical points at infinity of the associated variational problem and compute their contribution of the difference of topology between the level sets of the associated Euler-Lagrange functional. We then use topological and dynamical methods to prove some existence and multiplicity results.

On a quasilinear mean field equation with an exponential nonlinearity

“…The non-contractibility of B m (Ω) let us see a change in topology between J L λ and J −L λ for L > 0 large, and by Lemma 7 we obtain the existence result claimed in Theorem 3. Notice that our approach is simpler than the one in [33][34][35] (see also [9]), by using [53] instead of the Struwe's monotonicity trick to bypass the general failure of PS-condition for J λ . As already explained, the key point is the following improvement of the Moser-Trudinger inequality:…”