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.