Existence and non-existence of Schwarz symmetric ground states for elliptic eigenvalue problems

We determine a class of Carathéodory functions G for which the minimum formulated in the problem (1.1) below is achieved at a Schwarz symmetric function satisfying the constraint. Our hypotheses about G seem natural and, as our examples show, they are optimal from some points of vie

Symmetrization Inequalities for Composition Operators of Carathéodory Type

Let F:(0, ∞) × [0, ∞) → R be a function of Carathéodory type. We establish the inequality $$\int_{\mathbb{R}^{N}} F( | x |, u(x) ) dx \leq \int_{\mathbb{R}^{N} } F( | x |, u^{\ast}(x)) dx.$$ where u* denotes the Schwarz symmetrization of u, under hypotheses on F that seem quite natural when this inequality is used to obtain existence results in the context of elliptic partial differential equations. We also treat the case where RN is replaced by a set of finite measure. The identity $$\int_{\mathbb{R}^{N}} G(u(x)) dx = \int_{\mathbb{R}^{N}} G(u^{\ast}(x)) dx$$ is also discussed under the assumption that G: [0,∞) → R is a Borel function. 2000 Mathematics Subject Classification 26D20, 42C20, 46E3

Orbital stability: analysis meets geometry

We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the wave equation, and for the Manakov system

A guide to the Choquard equation

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$-\Delta u + V(x)u = \bigl(|x|^{-(N-\alpha)} * |u|^p\bigr)|u|^{p - 2} u \qquad \text{in $\mathbb{R}^N$},$$ and some of its variants and extensions.Comment: 39 page