PERTURBATION THEOREMS FOR FRACTIONAL CRITICAL EQUATIONS ON BOUNDED DOMAINS

We consider the fractional critical problem $A_{s}u=K(x)u^{(n+2s)/(n-2s)},u>0$ in $\unicode[STIX]{x1D6FA},u=0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ , where $A_{s},s\in (0,1)$ , is the fractional Laplace operator and $K$ is a given function on a bounded domain $\unicode[STIX]{x1D6FA}$ of $\mathbb{R}^{n},n\geq 2$ . This is based on A. Bahri’s theory of critical points at infinity in Bahri [Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182 (Longman Scientific & Technical, Harlow, 1989)]. We prove Bahri’s estimates in the fractional setting and we provide existence theorems for the problem when $K$ is close to 1.

Prescribing Morse Scalar Curvatures: Blow-Up Analysis

We study finite-energy blow-ups for prescribed Morse scalar curvatures in both the subcritical and the critical regime. After general considerations on Palais–Smale sequences, we determine precise blow-up rates for subcritical solutions: in particular the possibility of tower bubbles is excluded in all dimensions. In subsequent papers, we aim to establish the sharpness of this result, proving a converse existence statement, together with a one-to-one correspondence of blowing-up subcritical solutions and critical points at infinity. This analysis will be then applied to deduce new existence results for the geometric problem.

β-flatness condition in CR spheres multiplicity results

We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy–Riemann spheres under [Formula: see text]-flatness condition. To find a lower bound for the number of solutions, we use Bahri’s methods based on the theory of critical points at infinity and a Poincaré–Hopf-type formula.

Global Phase Portrait and Bifurcation Diagram for a Multi-Parametric Linear System

The goal of this article is to conduct a global dynamics study of a linear multiparameter system (real parameters (a,b,c) in R^3); for this, we take the different changes that these parameters present. First, we find the different parametric surfaces in which the space is divided, where the stability of the critical point is defined; we then create a bifurcation diagram to classify the different bifurcations that appear in the system. Finally, we determine and classify the critical points at infinity, considering the canonical shape of the Poincaré sphere, and thus, obtain a global phase portrait of the multiparametric linear system.

Prescribing scalar curvatures: non compactness versus critical points at infinity

We illustrate an example of a generic, positive function K on a Riemannian manifold to be conformally prescribed as the scalar curvature, for which the corresponding Yamabe type L2-gradient flow exhibits non compact flow lines, while a slight modification of it is compact.

Theory of “Critical Points at Infinity” and a Resonant Singular Liouville-Type Equation

We consider the following Liouville-type equation on domains ofwhereUsing some dynamical and topological tools from the “critical point theory at infinity” of Bahri, we study the critical points at infinity of the related variational problem. Then we derive from our analysis some existence results in the so-called resonant case, that is, when the parameter ϱ is of the form