Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.57 (2025)

In this paper we study autonomous systems whose Lagrangian function is the combination of several homogeneous terms with respect to positions and velocities. We show that, assuming certain relations between the degrees of homogeneity of such terms, the systems considered possess (in addition to energy) a further first integral that provides information about their solutions. A new feature of these results is the use of the theory of nonlocal constants, which finds useful constants using one-parameter perturbed motions.

By using mountain pass arguments and a novel grouptheoretical approach, in this paper we study the existence of multiple sequences of nodal solutions with prescribed different symmetries for a wide class of (sub)critical elliptic problems settled on the unit sphere (Sd, h), of dimension d ≥ 5, whose simple prototype is given by the celebrated Yamabe equation on the sphere.

In this article, we introduce the concept of generalized (α, T)-G-contractive mappings in partial b-metric spaces endowed with a digraph G and obtain a new coincidence point and common fixed point result for a pair of self mappings satisfying such contractive condition. Our main result will extend and unify several known results in the existing literature and also brings some new results as consequences. Finally, we give an application of our main result to obtain a unique solution of an integral equation.

In this paper, we produce explicit examples of mean curvature flow of (2m − 1)-dimensional submanifolds which converge to (2m − 2)-dimensional submanifolds at a finite time. These examples are a special class of hyperspheres in Cm with a U(m)-invariant Kähler metrics. We first discuss the mean curvature flow problem and then investigate the type of singularities for them.