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Denis Bonheure

Publications and source records attributed to Denis Bonheure.

At least 37 records · Page 2Linked to original sources
arXiv2018

Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation $$ γΔ^2 u -Δu + αu=|u|^{2 σ} u, \quad u \in H^2(\R^N), $$ under the constraint $$ \int_{\R^N}|u|^2 \, dx =c>0. $$ We assume $γ>0, N \geq 1, 4 \leq σN < \frac{4N}{(N-4)^+}$, whereas the parameter $α\in \R$ will appear as a Lagrange multiplier. Given $c \in \R^+$, we consider several questions including the existence of ground states, of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.

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arXiv2018

Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation

We study the mixed dispersion fourth order nonlinear Schrödinger equation \begin{equation*} %\tag{\protect{4NLS}}\label{4nls} i \partial_t ψ-γΔ^2 ψ+βΔψ+|ψ|^{2σ} ψ=0\ \text{in}\ \R \times\R^N, \end{equation*} where $γ,σ>0$ and $β\in \R$. We focus on standing wave solutions, namely solutions of the form $ψ(x,t)=e^{iαt}u(x)$, for some $α\in \R$. This ansatz yields the fourth-order elliptic equation \begin{equation*} %\tag{\protect{*}}\label{4nlsstar} γΔ^2 u -βΔu +αu =|u|^{2σ} u. \end{equation*} We consider two associated constrained minimization problems: one with a constraint on the $L^2$-norm and the other on the $L^{2σ+2}$-norm. Under suitable conditions, we establish existence of minimizers and we investigate their qualitative properties, namely their sign, symmetry and decay at infinity as well as their uniqueness, nondegeneracy and orbital stability.

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arXiv2018

Singular radial solutions for the Keller-Segel equation in high dimension

We study singular radially symmetric solution of the stationary Keller-Segel equation, that is, an elliptic equation with exponential nonlinearity, which is super-critical in dimension $N \geq 3$. The solutions are unbounded at the origin and we show that they describe the asymptotics of bifurcation branches of regular solutions. It is shown that for any ball and any $k \geq 0$, there is a singular solution that satisfies Neumann boundary condition and oscillates at least $k$ times around the constant equilibrium. Moreover, we prove that in dimension $3 \leq N \leq 9$ there are regular solutions satisfying Neumann boundary conditions that are close to singular ones. Hence, it follows that there exist regular solutions on any ball with arbitrarily fast oscillations. For generic radii, we show that the bifurcation branches of regular solutions oscillate in the bifurcation plane when $3\leq N\leq 9$ and approach to a singular solution. In dimension $N > 10$, we show that the Morse index of the singular solution is finite, and therefore the existence of regular solutions with fast oscillations is not expected.

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arXiv2018

On the regularity of the minimizer of the electrostatic Born-Infeld energy

We consider the electrostatic Born-Infeld energy \begin{equation*} \int_{\mathbb{R}^N}\left(1-{\sqrt{1-|\nabla u|^2}}\right)\, dx -\int_{\mathbb{R}^N}ρu\, dx, \end{equation*} where $ρ\in L^{m}(\mathbb{R}^N)$ is an assigned charge density, $m \in [1,2_*]$, $2_*:=\frac{2N}{N+2}$, $N\geq 3$. We prove that if $ρ\in L^q(\mathbb{R}^N) $ for $q>2N$, the unique minimizer $u_ρ$ is of class $W_{loc}^{2,2}(\mathbb{R}^N)$. Moreover, if the norm of $ρ$ is sufficiently small, the minimizer is a weak solution of the associated PDE \begin{equation}\label{eq:BI-abs} \tag{$\mathcal{BI}$} -\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= ρ\quad\hbox{in }\mathbb{R}^N, \end{equation} with the boundary condition $\lim_{|x|\to\infty}u(x)=0$ and it is of class $C^{1,α}_{loc}(\mathbb{R}^N)$, for some $α\in (0,1)$.

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arXiv2018

On a fourth order nonlinear Helmholtz equation

In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz equation $Δ^2 u -βΔu + αu= Γ|u|^{p-2} u$ in $\mathbb R^N$ for positive, bounded and $\mathbb Z^N$-periodic functions $Γ$. Using the dual method of Evequoz and Weth, we find solutions to this equation and establish some of their qualitative properties.

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arXiv2017

On the Born-Infeld equation for electrostatic fields with a superposition of point charges

In this paper, we study the static Born-Infeld equation $$ -\mathrm{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)=\sum_{k=1}^n a_kδ_{x_k}\quad\mbox{in }\mathbb R^N,\qquad \lim_{|x|\to\infty}u(x)=0, $$ where $N\ge3$, $a_k\in\mathbb R$ for all $k=1,\dots,n$, $x_k\in\mathbb R^N$ are the positions of the point charges, possibly non symmetrically distributed, and $δ_{x_k}$ is the Dirac delta distribution centered at $x_k$. For this problem, we give explicit quantitative sufficient conditions on $a_k$ and $x_k$ to guarantee that the minimizer of the energy functional associated to the problem solves the associated Euler-Lagrange equation. Furthermore, we provide a more rigorous proof of some previous results on the nature of the singularities of the minimizer at the points $x_k$'s depending on the sign of charges $a_k$'s. For every $m\in\mathbb N$, we also consider the approximated problem $$ -\sum_{h=1}^mα_hΔ_{2h}u=\sum_{k=1}^n a_kδ_{x_k}\quad\mbox{in }\mathbb R^N, \qquad\lim_{|x|\to\infty}u(x)=0 $$ where the differential operator is replaced by its Taylor expansion of order $2m$, see (2.1). It is known that each of these problems has a unique solution. We study the regularity of the approximating solution, the nature of its singularities, and the asymptotic behavior of the solution and of its gradient near the singularities.

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arXiv2017

Strong instability of ground states to a fourth order Schrödinger equation

In this note we prove the instability by blow-up of the ground state solutions for a class of fourth order Schr\" odinger equations. This extends the first rigorous results on blowing-up solutions for the biharmonic NLS due to Boulenger and Lenzmann \cite{BoLe} and confirm numerical conjectures from \cite{BaFi, BaFiMa1, BaFiMa, FiIlPa}.

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arXiv2016

The logarithmic Choquard equation: sharp asymptotics and nondegeneracy of the groundstate

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation $$ - Δu + a u = \frac{1}{2 π} \Bigl[\ln \frac{1}{|x|}* |u|^2 \Bigr] \ u \qquad \text{in $\mathbb{R}^2$} $$ and we establish its nondegeneracy. For the corresponding three-dimensional problem, the nondegeneracy property of the positive ground state to the Choquard equation was proved by E. Lenzmann (Analysis & PDE, 2009).

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arXiv2016

A quasilinear bistable equation in cylinders and timelike heteroclinics in special relativity

In this note we consider the action functional \[ \int_{\mathbb{R} \times ω} \left( 1 - \sqrt{ 1 - |\nabla u|^2 } + W(u) \right) \, \mathrm{d}t, \] where $W$ is a double well potential and $ω$ is a bounded domain of $\mathbb{R}^{N-1}$. We prove existence, one-dimensionality and uniqueness (up to translation) of a smooth minimizing phase transition between the two stable states $u=1$ and $u=-1$. The question of existence of at least one minimal heteroclinic connection for the non autonomous model \[ \int_{\mathbb{R}} \left( 1 - \sqrt{1-|u'|^2} + a(t) W(u) \right) \, \mathrm{d}t \] is also addressed. For this, we look for the possible assumptions on $a(t)$ ensuring the existence of a minimizer.

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arXiv2016

Paths to uniqueness of critical points and applications to partial differential equations

We prove a unified and general criterion for the uniqueness of critical points of a functional in the presence of constraints such as positivity, boundedness, or fixed mass. Our method relies on convexity properties along suitable paths and significantly generalizes well-known uniqueness theorems. Due to the flexibility in the construction of the paths, our approach does not depend on the convexity of the domain and can be used to prove uniqueness in subsets, even if it does not hold globally. The results apply to all critical points and not only to minimizers, thus they provide uniqueness of solutions to the corresponding Euler-Lagrange equations. For functionals emerging from elliptic problems, the assumptions of our abstract theorems follow from maximum principles, decay properties, and novel general inequalities. To illustrate our method we present a unified proof of known results, as well as new theorems for mean-curvature type operators, fractional Laplacians, Hamiltonian systems, Schrödinger equations, and Gross-Pitaevski systems.

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arXiv2016

Properties of groundstates of nonlinear Schrödinger equations under a weak constant magnetic field

We study the qualitative properties of groundstates of the time-independent magnetic semilinear Schrödinger equation \[ - (\nabla + i A)^2 u + u = |u|^{p-2} u, \qquad \text{ in } \mathbb{R}^N, \] where the magnetic potential $A$ induces a constant magnetic field. When the latter magnetic field is small enough, we show that the groundstate solution is unique up to magnetic translations and rotations in the complex phase space, that groundstate solutions share the rotational invariance of the magnetic field and that the presence of a magnetic field induces a Gaussian decay. In this small magnetic field régime, the corresponding ground-energy is a convex differentiable function of the magnetic field.

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arXiv2016

Multiple positive solutions of the stationary Keller-Segel system

We consider the stationary Keller-Segel equation \begin{equation*} \begin{cases} -Δv+v=λe^v, \quad v>0 \quad & \text{in }Ω,\\ \partial_νv=0 &\text{on } \partial Ω, \end{cases} \end{equation*} where $Ω$ is a ball. In the regime $λ\to 0$, we study the radial bifurcations and we construct radial solutions by a gluing variational method. For any given natural positive number $n$, we build a solution having multiple layers at $r_1,\ldots,r_n$ by which we mean that the solutions concentrate on the spheres of radii $r_i$ as $λ\to 0$ (for all $i=1,\ldots,n$). A remarkable fact is that, in opposition to previous known results, the layers of the solutions do not accumulate to the boundary of $Ω$ as $λ\to 0$. Instead they satisfy an optimal partition problem in the limit.

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arXiv2016

Qualitative properties of solutions to mixed-diffusion bistable equations

We consider a fourth-order extension of the Allen-Cahn model with mixed-diffusion and Navier boundary conditions. Using variational and bifurcation methods, we prove results on existence, uniqueness, positivity, stability, a priori estimates, and symmetry of solutions. As an application, we construct a nontrivial bounded saddle solution in the plane.

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arXiv2016

Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

Assuming $B_{R}$ is a ball in $\mathbb R^{N}$, we analyze the positive solutions of the problem \[ \begin{cases} -Δu+u= |u|^{p-2}u, &\text{ in } B_{R},\newline \partial_νu=0,&\text{ on } \partial B_{R}, \end{cases} \] that branch out from the constant solution $u=1$ as $p$ grows from $2$ to $+\infty$. The non-zero constant positive solution is the unique positive solution for $p$ close to $2$. We show that there exist arbitrarily many positive solutions as $p\to\infty$ (in particular, for supercritical exponents) or as $R \to \infty$ for any fixed value of $p>2$, answering partially a conjecture in [Bonheure-Noris-Weth]. We give the explicit lower bounds for $p$ and $R$ so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.

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arXiv2016

One-dimensional symmetry and Liouville type results for the fourth order Allen-Cahn equation in R$^N$

In this paper, we prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R N , as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. We also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.

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arXiv2015

On the electrostatic Born-Infeld equation with extended charges

In this paper, we deal with the electrostatic Born-Infeld equation \begin{equation}\label{eq:BI-abs} \tag{$\mathcal{BI}$} \left\{ \begin{array}{ll} -\operatorname{div}\left(\displaystyle\frac{\nabla ϕ}{\sqrt{1-|\nabla ϕ|^2}}\right)= ρ, & \hbox{in } \mathbb{R}^N, \\ \displaystyle\lim_{|x|\to \infty}ϕ(x)= 0, \end{array} \right. \end{equation} where $ρ$ is an assigned extended charge density. We are interested in the existence and uniqueness of the potential $ϕ$ and finiteness of the energy of the electrostatic field $-\nabla ϕ$. We first relax the problem and treat it with the direct method of the Calculus of Variations for a broad class of charge densities. Assuming $ρ$ is radially distributed, we recover the weak formulation of \eqref{eq:BI-abs} and the regularity of the solution of the Poisson equation (under the same smootheness assumptions). In the case of a locally bounded charge, we also recover the weak formulation without assuming any symmetry. The solution is even classical if $ρ$ is smooth. Then we analyze the case where the density $ρ$ is a superposition of point charges and discuss the results in [Kiessling, Comm. Math. Phys. 314 (2012), 509--523]. Other models are discussed, as for instance a system arising from the coupling of the nonlinear Klein-Gordon equation with the Born-Infeld theory.

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arXiv2015

Nonlinear Schr{ö}dinger equation: concentration on circles driven by an external magnetic field

In this paper, we study the semiclassical limit for the stationary magnetic nonlinear Schrödinger equation \begin{align}\label{eq:initialabstract}\left( i \hbar \nabla + A(x) \right)^2 u + V(x) u = |u|^{p-2} u, \quad x\in \mathbb{R}^{3},\end{align}where $p\textgreater{}2$, $A$ is a vector potential associated to a given magnetic field $B$, i.e $\nabla \times A =B$ and $V$ is a nonnegative, scalar (electric) potential which can be singular at the origin and vanish at infinity or outside a compact set.We assume that $A$ and $V$ satisfy a cylindrical symmetry. By a refined penalization argument, we prove the existence of semiclassical cylindrically symmetric solutions of upper equation whose moduli concentrate, as $\hbar \to 0$, around a circle. We emphasize that the concentration is driven by the magnetic and the electric potentials. Our result thus shows that in the semiclassical limit, the magnetic field also influences the location of the solutions of $(\ref{eq:initialabstract})$ if their concentration occurs around a locus, not a single point.

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