Christopher Grumiau, Marco Squassina, Christophe Troestler
We discuss the application of the Mountain Pass algorithm to the so-called quasi-linear Schrodinger equation, which is naturally associated with a class of nonsmooth functionals so that the classical algorithm is not directly applicable.
C. Grumiau, M. Squassina, C. Troestler
In this paper, we study equations driven by a non-local integrodifferential operator $\mathcal{L}_K$ with homogeneous Dirichlet boundary conditions. More precisely, we study the problem \[ \begin{aligned} &- \mathcal{L}_K u + V(x)u = |u|^{p-2}u, &&\text{in } Ω, \newline &u=0, &&\text{in } \mathbb{R}^N \setminus Ω, \end{aligned} \] where $2 < p < 2^{*}_s = \frac{2N}{N-2s}$, $Ω$ is an open bounded domain in $\mathbb{R}^{N}$ for $N\ge 2$ and $V$ is a $L^\infty$ potential such that $-\mathcal{L}_K + V$ is positive definite. As a particular case, we study the problem \[ \begin{aligned} &(- Δ)^s u + V(x)u = |u|^{p-2}u, &&\text{in } Ω, \newline &u=0, &&\text{in } \mathbb{R}^N \setminus Ω, \end{aligned} \] where $(-Δ)^s$ denotes the fractional Laplacian (with $0<s<1$). We give assumptions on $V$, $Ω$ and $K$ such that ground state solutions (resp. least energy nodal solutions) respect the symmetries of some first (resp. second) eigenfunctions of $-\mathcal{L}_K + V$, at least for $p$ close to $2$. We study the uniqueness, up to a multiplicative factor, of those types of solutions. The results extend those obtained for the local case.
Marco Caliari, Marco Squassina
We provide some numerical computations for the soliton dynamics of the nonlinear Schrödinger equation with an external potential. After computing the ground state solution $r$ of a related elliptic equation we show that, in the semi-classical regime, the center of mass of the solution with initial datum modelled on $r$ is driven by the solution of a Newtonian type law. Finally, we provide some examples and analyze the numerical errors in the two dimensional case when $V$ is an harmonic potential.