Introduction v Chapter 1. The Geometric setting 1.1. Cut-locus and volume growth function 1.2. Model manifolds and basic comparisons 1.3. Some spectral theory on manifolds Chapter 2. Some geometric examples related to oscillation theory 2.1. Conjugate points and Myers type compactness results 2.2. The spectrum of the Laplacian on complete manifolds 2.3. Spectral estimates and immersions 2.4. Spectral estimates and nonlinear PDE Chapter 3. On the solutions of the ODE (vz) + Avz = 0 3.1. Existence, uniqueness and the behaviour of zeroes 3.2. The critical curve: definition and main estimates Chapter 4. Below the critical curve 4.1. Positivity and estimates from below 4.2. Stability, index of −∆ − q(x) and the uncertainty principle 4.3. A comparison at infinity for nonlinear PDE 4.4. Yamabe type equations with a sign-changing nonlinearity 4.5. Upper bounds for the number of zeroes of z Chapter 5. Exceeding the critical curve 5.1. First zero and oscillation 5.2. Comparison with known criteria 5.3. Instability and index of −∆ − q(x) 5.4. Some remarks on minimal surfaces 5.5. Newton operators, unstable hypersurfaces and the Gauss map 5.6. Dealing with a possibly negative potential 5.7. An extension of Calabi compactness criterion Chapter 6. Much above the critical curve 6.1. Controlling the oscillation 6.2. The growth of the index of −∆ − q(x) 6.3. The essential spectrum of −∆ and punctured manifolds Bibliography

Set in Riemannian enviroment, the aim of this paper is to present and discuss some equivalent characterizations of the Liouville property relative to special operators, in some sense modeled after the p-Laplacian with potential. In particular, we discuss the equivalence between the Lioville property and the Khas'minskii condition, i.e. the existence of an exhaustion functions which is also a supersolution for the operator outside a compact set. This generalizes a previous result obtained by one of the authors and answers to a question in [26].Date: September 10, 2018. 2010 Mathematics Subject Classification. Primary 31C12 (potential theory on Riemannian manifolds, Secondary: 35B53 (Liouville theorems), 58J65 (stochastic equations and processes on manifolds), 58J05 (elliptic equations on manifolds).

In this paper we obtain generalized Keller-Osserman conditions for wide classes of differential inequalities on weighted Riemannian manifolds of the formPrototypical examples of these operators are the p-Laplacian and the mean curvature operator. While we concentrate on non-existence results, in many instances the conditions we describe are in fact necessary for non-existence. The geometry of the underlying manifold does not affect the form of the Keller-Osserman conditions, but is reflected, via bounds for the modified Bakry-Emery Ricci curvature, by growth conditions for the functions b and ℓ. We also describe a weak maximum principle related to inequalities of the above form which extends and improves previous results valid for the ϕ-Laplacian. F (t)∈ L 1 (+∞),2000 Mathematics Subject Classification. 58J05, 35J60.

Our work proposes a unified approach to three different topics in a general Riemannian setting: splitting theorems, symmetry results and overdetermined elliptic problems. By the existence of a stable solution to the semilinear equation −∆u = f (u) on a Riemannian manifold with non-negative Ricci curvature, we are able to classify both the solution and the manifold. We also discuss the classification of monotone (with respect to the direction of some Killing vector field) solutions, in the spirit of a conjecture of De Giorgi, and the rigidity features for overdetermined elliptic problems on submanifolds with boundary. R ) = o(R 2 log R) as R → +∞. Remark 6. For R 2 , the above theorem generalizes the one-dimensional symmetry result in Theorem 1.2 of [18]. See also [37] for interesting studies on the geometric and topological properties of overdetermined problems in the Euclidean plane.By the monotonicity Theorem 1.1 in [4], the relation ∇u, X > 0 on Ω is authomatic for globally Lipschitz epigraphs Ω of Euclidean space and for some large class of nonlinearities f including the prototype Allen-Cahn one f (u) = u − u 3 (even without requiring the Neumann condition in (6)). However, it is an open problem to enlarge the class of domains Ω ⊆ R m for which ∇u, X > 0 is met, or to find nontrivial analogues on Riemannian manifolds. In the last section, we move some steps towards this problem by proving some lemmata that may have independent interest. In particular, we obtain the next result:Proposition 7. Let (M, , ) be a complete Riemannian manifold satisfying Ric ≥ −(m − 1)H 2 , , for some H ≥ 0, and let f ∈ C 1 (R) have the propertiesfor some λ > 0 and some small δ 0 , s 0 > 0. Let Ω ⊆ M be an open, connected subset, and suppose that u ∈ C 2 (Ω) ∩ C 0 (Ω) is a bounded, non-negative solution of −∆u = f (u) on Ω,

We study the qualitative behavior of non-negative entire solutions of differential inequalities with gradient terms on the Heisenberg group. We focus on two classes of inequalities: ϕ u f (u)l(|∇u|)continuous functions satisfying certain monotonicity properties.The operator ϕ , called the ϕ-Laplacian, generalizes the p-Laplace operator considered by various authors in this setting. We prove some Liouville theorems introducing two new Keller-Osserman type conditions, both extending the classical one which appeared long ago in the study of the prototype differential inequality u f (u) in R m . We show sharpness of our conditions when we specialize to the p-Laplacian. While proving these results we obtain a strong maximum principle for ϕ which, to the best of our knowledge, seems to be new. Our results continue to hold, with the obvious minor modifications, also for Euclidean space.