Eugenio Massa

On a variational characterization of the Fučík spectrum of the Laplacian and a superlinear Sturm–Liouville equation

In the first part of this paper, a variational characterization of parts of the Fucik spectrum for the Laplacian in a bounded domain Ω is given. The proof uses a linking theorem on sets obtained through a suitable deformation of subspaces of H 1 (Ω). In the second part, a nonlinear Sturm–Liouville equation with Neumann boundary conditions on an interval is considered, where the nonlinearity intersects all but a finite number of eigenvalues. It is proved that, under certain conditions, this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values related to the Fucik spectrum.

Superlinear elliptic problems with sign changing coefficients

Via variational methods, we study multiplicity of solutions for the problem where a simple example for g(x, u) is |u|p-2u; here a, λ are real parameters, 1 0, and a total of five nontrivial solutions are obtained when λ is small and a ≥ λ1. Note that this type of results are valid even in the critical case.

Positive definite functions on complex spheres and their walks through dimensions

We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We show that the analogues of Montee and Descente operators as proposed by Beatson and zu Castell [J. Approx. Theory 221 (2017), 22-37] on the basis of the original Matheron operator [Les variables regionalisees et leur estimation, Masson, Paris, 1965], allow for similar walks through dimensions. We show that the Montee operators also preserve, up to a constant, strict positive definiteness. For the Descente operators, we show that strict positive definiteness is preserved under some additional conditions, but we provide counterexamples showing that this is not true in general. We also provide a list of parametric families of (strictly) positive definite functions over complex spheres, which are important for several applications.

On necessary conditions for the comparison principle and the sub- and supersolution method for the stationary Kirchhoff equation

In this paper, we propose a counterexample to the validity of the comparison principle and of the sub- and supersolution method for nonlocal problems like the stationary Kirchhoff equation. This counterexample shows that in general smooth bounded domains in any dimension, these properties cannot hold true if the nonlinear nonlocal term M(u2) is somewhere increasing with respect to the H01-norm of the solution. Comparing with the existing results, this fills a gap between known conditions on M that guarantee or prevent these properties and leads to a condition that is necessary and sufficient for the validity of the comparison principle. It is worth noting that equations similar to the one considered here have gained interest recently for appearing in models of thermo-convective flows of non-Newtonian fluids or of electrorheological fluids, among others.