Silvia Cingolani (Università degli Studi di Bari Aldo Moro).
Título: Concentration phenomena for a fractional problem without Nehari constraint.
Abstract: The effects of the topology on the number of positive solutions for Dirichlet boundary value problems go back to celebrated papers due to Bahri-Coron and Benci-Cerami. I will present some multiplicity results for local and nonlocal problems without Nehari constraint. Some delicate aspects arise in presence of nonlocal operators and, in particular, a new good notion of center mass is needed. The results are contained in some joint papers with Louis Jeanjean (University Franche-Compté, Besancon), Kazunaga Tanaka (Waseda University, Tokyo) and in a recent paper with Marco Gallo (University of Bari).
Salvador López Martínez (INRIA Lille - Nord Europe).
Título: The method of Gidas and Spruck in elliptic problems with natural growth in the gradient.
Abstract: A classical problem in the literature consists of the existence of nontrivial solutions to semilinear Dirichlet problems with a power-like nonlinearity with superlinear growth. An outstanding contribution to this problem was the celebrated work of Gidas and Spruck, in which the authors considered the so-called subcritical case and proved a priori estimates of the solutions via a blow-up method in a rather general setting. On the other hand, the study of elliptic problems with natural growth in the gradient was initiated by the works of Boccardo, Murat and Puel and this topic has by now become classical too. In this talk, we analyze an elliptic problem which may be seen as a combination of both previous classical problems. We will show that the blow-up method due to Gidas and Spruck can be adapted in a nontrivial way to this combined problem. Our approach has the advantage of working in a general framework as non-constant coefficients of the gradient term and singularities are allowed.