RESEARCH TOPICS:
Conformal Geometry, especially curvature prescription problems and conformally invariant equations (Q-curvature);
Nonlinear and nonlocal elliptic PDEs, with emphasis on existence, compactness and qualitative properties of solutions, blow-up analysis and concentration phenomena, particularly in problems with critical or exponential growth. I also dealt with Choquard-type equations and Mean-Field Games models with Choquard-type interaction;
Overdetermined boundary value problems, including Serrin-type and p-Laplacian problems;
Variational methods, such as min-max techniques, critical point theory and perturbative arguments;
Publications:
Bernardini C. : Existence and Compactness of Conformal Metrics on the Plane with Unbounded and Sign-Changing Gaussian Curvature, Vietnam J. Math. (2021). DOI:10.1007/s10013-021-00540-5 2 (preprint Arxiv: 2304.14305).
Bernardini C. : Existence and asymptotic behavior of non-normal conformal metrics on R⁴ with sign-changing Q-curvature, Commun. Contemp. Math. 25 (2023), no. 10, Paper No. 2250053. DOI:10.1142/S0219199722500535 (preprint ArXiv: 2112.10848).
Bernardini C., Vespri V., Zaccaron M. : A note on Campanato’s L^p -regularity with continuous coefficients, Eurasian Math. J. 13 (2022) no. 4 , 44 – 53. DOI:10.32523/2077-9879-2022-13-4-44-53.
Bernardini C., Cesaroni A.: Ergodic Mean-Field Games with aggregation of Choquard-type, J. Differential Equations 364 (2023) 296-335. DOI:10.1016/j.jde.2023.03.045 (preprint ArXiv: 2208.08177).
Bernardini C.: Mass concentration for Ergodic Choquard Mean-Field Games, ESAIM: COCV Vol.31 (2025) DOI: 10.1051/cocv/2023083 (preprint ArXiv:2212.00132).
Bernardini C., Cesaroni A. : Boundary value problems for Choquard equations, Nonlinear Analysis 254 (2025), Paper No.113745. DOI:10.1016/j.na.2024.113745 (preprint Arxiv: 2305.09043)
Agostiniani V., Bernardini C., Borghini S., Mazzieri L.: On the Serrin problem for ring-shaped domains: the n-dimensional case, The Journal of Geometric Analysis (2026) DOI: 10.1007/s12220-026-02390-4 (preprint ArXiv: 2505.04458)
Preprints: