CLASSIFICATION RESULTS FOR SEMILINEAR ELLIPTIC EQUATIONS

This thesis addresses classification problems of solutions to semilinear elliptic equations, with a particular focus on the study of symmetry and rigidity results. The thesis is divided into four chapters. In Chapter 1 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. In Chapter 2 we consider a family of critical elliptic equations which arise as the Euler-Lagrange equation of Caffarelli-Kohn-Nirenberg inequalities, possibly in convex cones in R^N, with N bigger or equal than 2. We classify positive solutions without assuming energy assumptions on the solution and when the intrinsic dimension n in (3/2,5]. These results will follow as an application of the approach introduced in Chapter 1. Chapter 3 concerns two main topics. The first one is about semilinear elliptic equations with mixed boundary conditions in spherical sectors inside a cone. We show that a radial symmetry result of Gidas-Ni-Nirenberg type for positive solutions does not hold in general nonconvex cones. This symmetry breaking result is achieved by studying the Morse index of radial positive solutions and analyzing how it depends on the domain D on the unit sphere which spans the cone. In particular, it is proved that the Neumann eigenvalues of the Laplace Beltrami operator on D play a role in computing the Morse index. The second one is about a similar breaking of symmetry result, obtained for positive solutions of the critical Neumann problem in the whole unbounded cone. In this case, it is proved that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones. Chapter 4 is about a quantitative Sobolev inequality in cones satisfying the property that Aubin-Talenti bubbles are the only minimizers (for instance convex cones). We follow a Bianchi-Egnell approach to obtain a quantitative estimate in terms of the first Neumann eigenvalue of the Laplace Beltrami operator on the set D on the unit sphere, which spans the cone.