Título: What is a subcritical nonlinearity?
Abstract: We consider a Dirichlet problem in a bounded smooth domain with a slightly subcritical nonlinearity of nonpower type. For this problem, standard compact embeddings cannot be used to guarantee the existence of a solution as in the case of power-type nonlinearities.
Specifically, when the nonlinearity is the critical power divided by a logarithm raised to a certain exponent, we provide a priori bounds and through the Leray-Schauder degree, we show existence of positive solutions. Our arguments rely on the moving planes method, a Pohozaev identity, gradient regularity results for q>N, and Morrey's Theorem. This technique is available in a certain range of the exponent, far away from zero and it is robust enough to be extended for the p-laplacian and for elliptic systems.
When the exponent goes to zero, we use a Ljapunov-Schmidt reduction method to show that there is a positive solution which concentrates at a non-degenerate critical point of the Robin function.
Those are joint results with Castro, Clapp, Damascelli, Mavinga, Pistoia, Saldaña, and Sanjuán.
Francesca de Marchis (Università La Sapienza di Roma).
Título: Critical points of the Moser-Trudinger functional.
Abstract: In 1971 Moser found the sharp constant for an inequality by Trudinger. Namely he proved that on any planar domain the functional F(u):= ∫ exp(u^2) is bounded if restricted to any sphere of H^1_0 of radius R, with R^2 ≤ 4pi, while if R^2>4pi the functional F is unbounded on the sphere of H^1_0 of radius R. On closed surfaces the same results hold, however in a work with A.Malchiodi, L.Martinazzi and P.D. Thizy we prove the existence of critical points of F constrained to any sphere of H^1.