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.
Christopher Grumiau, Christophe Troestler
For a functional $\E$ and a peak selection that picks up a global maximum of $\E$ on varying cones, we study the convergence up to a subsequence to a critical point of the sequence generated by a mountain pass type algorithm. Moreover, by carefully choosing stepsizes, we establish the convergence of the whole sequence under a "localization" assumption on the critical point. We illustrate our results with two problems: an indefinite Schrödinger equation and a superlinear Schrödinger system.
Robin Van Oirbeek, Christopher Grumiau, Tim Verdonck
The concordance probability or C-index is a popular measure to capture the discriminatory ability of a regression model. In this article, the definition of this measure is adapted to the specific needs of the frequency and severity model, typically used during the technical pricing of a non-life insurance product. Due to the typical large sample size of the frequency data in particular, two different adaptations of the estimation procedure of the concordance probability are presented. Note that the latter procedures can be applied to all different versions of the concordance probability.