Ricerc@Sapienza

Let (M,g) be a non-locally conformally flat compact Riemannian manifold with dimension N≥7. We are interested in finding positive solutions to the linear perturbation of the Yamabe problem
−Lgu+ϵu=uN+2N−2 in (M,g)

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 consider the Gelfand problem-Δu=ρ2V(x)euinΩu=0on∂Ω,where Ω is a planar domain and ρ is a positive small parameter. Under some conditions on the potential 0 < V∈ C∞(Ω ¯) , we provide the first examples of multiplicity for blowing-up solutions at a given point in Ω as ρ→ 0. The argument is based on a refined Lyapunov–Schmidt reduction and the computation of the degree of a finite-dimensional map.

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.

We consider the Lane-Emden problem in the unit ball B of ℝ^2 centered at the origin with Dirichlet boundary conditions and exponent ∈(1,+∞) of the power nonlinearity. We prove the existence of sign-changing solutions having 2 nodal domains, whose nodal line does not touch ∂ and which are non-radial. We call these solutions quasi-radial. The result is obtained for any p sufficiently large, considering least energy nodal solutions in spaces of functions invariant under suitable dihedral groups of symmetry and proving that they fulfill the required qualitative properties.