Positive Solutions of Some Nonlinear Elliptic Problems in Exterior Domains

In the seminar we will present the homonymous paper by Vieri Benci and Giovanna Cerami of 1987. In particular we will study positive solutions of an equation , an exterior domain. On these domains the problem does not admit any ground state solution and the Palais-Smale condition is not satisfied. Benci and Cerami crafted a compactness lemma showing that Palais-Smale sequences are ‘relatively compact up to bubbles’ (global solutions of the equation) and that the energy of the sequence splits in the limit (we get the energy of the solution plus the energy of the bubbles). Since the energy of the bubbles is bounded from below, we can find some levels in which the Palais-Smale condition is satisfied, hence we can use a deformation argument to find non-empty critical levels.