Marco Ghimenti

Nodal solutions for the Choquard equation

Abstract We consider the general Choquard equations − Δ u + u = ( I α ⁎ | u | p ) | u | p − 2 u where I α is a Riesz potential. We construct minimal action odd solutions for p ∈ ( N + α N , N + α N − 2 ) and minimal action nodal solutions for p ∈ ( 2 , N + α N − 2 ) . We introduce a new minimax principle for least action nodal solutions and we develop new concentration–compactness lemmas for sign-changing Palais–Smale sequences. The nonlinear Schrodinger equation, which is the nonlocal counterpart of the Choquard equation, does not have such solutions.

On the existence of the fundamental Eigenvalue of an elliptic problem in R N

Abstract We study an eigenvalue problem for functions in ℝN and we find sufficient conditions for the existence of the fundamental eigenvalue. This result can be applied to the study of the orbital stability of the standing waves of the nonlinear Schrödinger equation.

Least action nodal solutions for the quadratic Choquard equation

We prove the existence of a minimal action nodal solution for the quadratic Choquard equation

On the Dynamics of Solitons in the Nonlinear Schrödinger Equation

-\Delta u + u = \big(I_\alpha \ast |u|^2\big)u \quad\text{in }\; \mathbb R^N,

Semiclassical states for a static supercritical Klein–Gordon–Maxwell–Proca system on a closed Riemannian manifold

where

Geodesics in conical manifolds

I_\alpha

A multiplicity result for double singularly perturbed elliptic systems

is the Riesz potential of order

Semiclassical Limit For the Nonlinear Klein Gordon Equation in Bounded Domains

\alpha\in(0,N)