Existence of infinitely many radial and non-radial solutions for quasilinear Schrödinger equations with general nonlinearity

In this paper, we prove the existence of many solutions for the following quasilinear Schrödinger equation \begin{equation*} -\Delta u - u\Delta(|u|^2) + V(|x|)u = f(|x|,u),\qquad x \in \mathbb{R}^N. \end{equation*} Under some generalized assumptions on $f$, we obtain infinitely many radial solutions for $N\geq 2$, many non-radial solutions for $N=4$ and $N \geq 6$, and a non radial solution for $N=5$. Our results generalize and extend some existing results

Beginner's guide to Aggregation-Diffusion Equations

The aim of this survey is to serve as an introduction to the different techniques available in the broad field of Aggregation-Diffusion Equations. We aim to provide historical context, key literature, and main ideas in the field. We start by discussing the modelling and famous particular cases: Heat equation, Fokker-Plank, Porous medium, Keller-Segel, Chapman-Rubinstein-Schatzman, Newtonian vortex, Caffarelli-V\'azquez, McKean-Vlasov, Kuramoto, and one-layer neural networks. In Section 4 we present the well-posedness frameworks given as PDEs in Sobolev spaces, and gradient-flow in Wasserstein. Then we discuss the asymptotic behaviour in time, for which we need to understand minimisers of a free energy. We then present some numerical methods which have been developed. We conclude the paper mentioning some related problems

Numerical modeling of black holes as sources of gravitational waves in a nutshell

These notes summarize basic concepts underlying numerical relativity and in particular the numerical modeling of black hole dynamics as a source of gravitational waves. Main topics are the 3+1 decomposition of general relativity, the concept of a well-posed initial value problem, the construction of initial data for general relativity, trapped surfaces and gravitational waves. Also, a brief summary is given of recent progress regarding the numerical evolution of black hole binary systems.Comment: 28 pages, lectures given at winter school 'Conceptual and Numerical Challenges in Femto- and Peta-Scale Physics' in Schladming, Austria, 200