This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

We consider solutions to some semilinear elliptic equations on complete noncompact Riemannian manifolds and study their classification as well as the effect of their presence on the underlying manifold. When the Ricci curvature is non-negative, we prove both the classification of positive solutions to the critical equation and the rigidity for the ambient manifold. The same results are obtained when we consider solutions to the Liouville equation on Riemannian surfaces. The results are obtained via a suitable P-function whose constancy implies the classification of both the solutions and the underlying manifold. The analysis carried out on the P-function also makes it possible to classify non-negative solutions for subcritical equations on manifolds enjoying a Sobolev inequality and satisfying an integrability condition on the negative part of the Ricci curvature. Some of our results are new even in the Euclidean case. We apply the P-function approach also to a family of critical elliptic equations which arise as the Euler-Lagrange equation of Caffarelli-Kohn-Nirenberg inequalities, possibly in convex cones in RN, with N≥2. We classify positive solutions without assuming energy assumptions on the solution and when the intrinsic dimension n∈(3/2,5]. These results are obtained in joint works with G. Ciraolo and A. Farina. G. Ciraolo, A. Farina, C.C. Polvara, Classification results, rigidity theorems and semilinear PDEs on Riemannian manifolds: a P-function approach, arXiv:2406.13699,(2024). G. Ciraolo, C.C. Polvara, On the classification of extremals of Caffarelli-Kohn-Nirenberg inequalities, arXiv:2410.09478, (2024). This talk is part of the activity of the research Project : PRIN 2022 PNRR project 2022AKNSE4 "Variational and Analytical aspects of Geometric PDEs", founded by the European Union- Next Generation EU and it is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.