Ricerc@Sapienza

The Adimurthi–Druet inequality is an improvement of the standard Moser–Trudinger inequality by adding a L2-type perturbation, quantified by α∈[0, λ1), where λ1 is the first Dirichlet eigenvalue of the Laplacian on a smooth bounded domain. It is known that this inequality admits extremal functions, when the perturbation parameter α is small. By contrast, we prove here that the Adimurthi–Druet inequality does not admit any extremal, when the perturbation parameter αapproaches λ1.

We prove the existence of extremals for fractional Moser-Trudinger inequalities in an interval and on the whole real line. In both cases we use blow-up analysis for the corresponding Euler-Lagrange equation, which requires new sharp estimates obtained via commutator techniques.

In this work we study the existence of nodal solutions for the problem -Δu=λueu2+|u|pinΩ,u=0on∂Ω,where Ω ⊆ R2 is a bounded smooth domain and p→ 1 +. If Ω is a ball, it is known that the case p= 1 defines a critical threshold between the existence and the non-existence of radially symmetric sign-changing solutions. In this work we construct a blowing-up family of nodal solutions to such problem as p→ 1 +, when Ω is an arbitrary domain and λ is small enough.