TESI DI DOTTORATO "Variational and Geometric Analysis for Quasilinear Elliptic Equations and Systems"

In this thesis, we employ variational and geometric analysis to investigate quasilinear elliptic equations and systems arising from nonlinear phenomena with a significant environmental impact. We focus on quasilinear elliptic systems driven by various nonlinear operators, such as the p-Laplacian, and nonlinear sources that are allowed to exhibit both subcritical and critical growth. We aim to establish the existence of solutions for perturbation of specific eigenvalue problems, by employing variational and topological methods. To establish existence results for autonomous systems of quasilinear PDEs in the spirit of the celebrated paper by Amann and Zehnder, we develop a local Morse theory for functional associated to quasilinear elliptic systems. Our objective is to bridge relationship between the local behaviour of the functional near its critical points and differential notions like the Morse index. To this aim, by refining topological arguments introduced by Cingolani and Degiovanni in Banach product spaces, we establish the finiteness of the critical groups and we derive a Poincaré-Hopf formula in a Banach product space, in presence of both subcritical and critical nonlinear coupling. Furthermore, we address the lack of regularity in the associated Euler functional by employing a penalization method combined with uniform L^infty estimates. We also establish uniform boundedness results for anisotropic quasilinear systems, that are of interest within regularity theory. To show existence results for non-autonomous systems of quasilinear PDEs in the spirit of the celebrated paper of Landesman and Lazer, we consider the eigenvalue problem for quasilinear elliptic systems introduced by de Thélin. Leveraging the intrinsic geometric structure of the eigenvalue problem, we prove the simplicity and isolation of the first eigenvalue lambda_1. Furthermore, we show the existence of a sequence of eigenvalues by employing a suitable deformation lemma proved by Bonnet. Subsequently, we analyze a non-autonomous perturbation of the eigenvalue problem. In particular, the established spectral properties enable us to derive new sufficient Landesman-Lazer type conditions within the framework of quasilinear elliptic systems. While the quasilinear elliptic systems under consideration typically involve the p-Laplacian for 1 < p < N, where N denotes the dimension of the space R^N, we also investigate the geometric role of the N-Laplacian in the N-dimensional Euclidean Onofri inequality. This inequality was established by Del Pino and Dolbeault for smooth functions with compact support. Specifically, after extending this inequality to a suitable weighted Sobolev space via a density argument, we exploit its connection with the Liouville equation on R^N to prove an equivalence with the sharp logarithmic Moser-Trudinger inequality on the unit ball B_1 of R^N, implicitly contained in the work of Carleson and Chang. For the specific case N=2, we further highlight a geometric link to the unit sphere S^2 through the stereographic projection.