Denis Bonheure

• Prof. Dr.
• Professor at Université Libre de Bruxelles

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 e...

In this paper, we deal with the electrostatic Born–Infeld equation $$\left\{\begin{array}{ll}-\operatorname{div} \left(\displaystyle\frac{\nabla\phi}{\sqrt{1-|\nabla \phi|^2}} \right)= \rho \quad{in} \mathbb{R}^N, \\ \displaystyle\lim_{|x|\to \infty} \phi(x)= 0,\end{array}\right. \quad \quad \quad \quad ({\mathcal{BI}})$$where \({\rho}\) is an assi...

In this paper we prove existence of least energy nodal solutions for the Hamiltonian elliptic system with H\'enon-type weights \[ -\Delta u = |x|^{\beta} |v|^{q-1}v, \quad -\Delta v =|x|^{\alpha}|u|^{p-1}u\quad \mbox{ in } \Omega, \qquad u=v=0 \mbox{ on } \partial \Omega, \] where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, $N\geq 1$, $\...

We consider radial solutions of elliptic systems of the form ⎧ ⎨ ⎩ −u + u = a |x| f (u, v) in B R , −v + v = b |x| g(u, v) in B R , ∂ ν u = ∂ ν v = 0 o n ∂B R , where essentially a, b are assumed to be radially nondecreasing weights and f , g are nondecreasing in each component. With few assumptions on the nonlinearities, we prove the existence of...

We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrödinger equations of the form -ε2Δu+V(x)u = K(x)up, x ∈ ℝN, where V, K are positive continuous functions and p > 1 is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential V is allowed to vanish at infin...

We study the periodic motions of the coupled system $\mathscr S$, consisting of an incompressible Navier-Stokes fluid interacting with a structure formed by a rigid body subject to {\em undamped} elastic restoring forces and torque around its rotation axis. The motion of $\mathscr S$ is driven by the uniform flow of the liquid, far away from the bo...

We prove existence of time-periodic weak solutions to the coupled liquid-structure problem constituted by an incompressible Navier–Stokes fluid interacting with a rigid body of finite size, subject to an undamped linear restoring force. The fluid flow is generated by a uniform, time-periodic velocity field {\boldsymbol V} far from the body. We emph...

We furnish necessary and sufficient conditions for the occurrence of a Hopf bifurcation in a particularly significant fluid-structure problem, where a Navier-Stokes liquid interacts with a rigid body that is subject to an undamped elastic restoring force. The motion of the coupled system is driven by a uniform flow at spatial infinity, with constan...

We study certain significant properties of the equilibrium configurations of a rigid body subject to an undamped elastic restoring force, in the stream of a viscous liquid in an unbounded 3D domain. The motion of the coupled system is driven by a uniform flow at spatial infinity, with constant dimensionless velocity $\lambda$. We show that if $\lam...

We study the long-time behavior of an elliptic rigid body which is allowed to vertically translate and rotate in a 2D unbounded channel under the action of a Poiseuille flow at large distances. The motion of the fluid is modelled by the incompressible Navier-Stokes equations, while the motion of the solid is described through Newton's laws. In addi...

We investigate the behaviour of radial solutions to the Lin–Ni–Takagi problem in the ball BR⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_R \subset \mathbb {R}^N...

We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely $− div (∇u / \sqrt{1 − |∇u|^ 2}) = ρ$ in $R^N$, where $N ≥ 3$. We first prove a new gradient estimate for classical solutions with smooth data ρ. As a consequence we obtain that the unique weak solution of the equation satis...

We investigate the behaviour of radial solutions to the Lin-Ni-Takagi problem in the ball $B_R \subset \mathbb{R}^N$ for $N \ge 3$: \begin{equation*} \left \{ \begin{aligned} - \triangle u_p + u_p & = |u_p|^{p-2}u_p & \textrm{ in } B_R, \\ \partial_\nu u_p & = 0 & \textrm{ on } \partial B_R, \end{aligned} \right. \end{equation*} when $p $ is close...

A partially hinged, partially free rectangular plate is considered, with the aim of addressing the possible unstable end behaviors of a suspension bridge subject to wind. This leads to a nonlinear plate evolution equation with a nonlocal stretching active in the spanwise direction. The wind-flow in the chordwise direction is modeled through a pisto...

We construct several families of radial solutions for the stationary Keller-Segel equation in the disk. The first family consists in solutions which blow up at the origin, as a parameter goes to zero, and concentrate on the boundary. The second is made of solutions which blow up at the origin and concentrate on an interior sphere, while the solutio...

In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation{−Δu−γ|x|2u=1|x|s|u|ps−2u in RN∖{0},u≥0, where N≥3, s∈[0,2), ps=2(N−s)N−2 and γ∈(−∞,(N−2)24). We extend results of E.N. Dancer, F. Gladiali, M. Grossi (2017) [12] where only the case s=0 is considered. The results specially rely on a careful analysis of...

We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely \begin{equation*} -\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= \rho \quad \hbox{in }\mathbb{R}^N, \end{equation*} where $N\geq 3$. We first prove a new gradient estimate for classical s...

Fluid flows around an obstacle generate vortices which, in turn, generate lift forces on the obstacle. Therefore, even in a perfectly symmetric framework equilibrium positions may be asymmetric. We show that this is not the case for a Poiseuille flow in an unbounded 2D channel, at least for small Reynolds number and flow rate. We consider both the...

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sp...

We consider nonlinear 2nd-order elliptic problems of the type $$\begin{align*} & -\Delta u=f(u)\ \textrm{in}\ \Omega, \qquad u=0\ \textrm{on}\ \partial \Omega, \end{align*}$$where $\Omega $ is an open $C^{1,1}$–domain in ${{\mathbb{R}}}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power...

In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation $$\begin{cases} -\Delta u-\displaystyle\frac \gamma{|x|^2}u=\displaystyle\frac{1}{|x|^s}|u|^{p_s-2}u & \text{ in } \mathbb{R}^N\setminus\{0\},\\ u\geq 0, & \end{cases}$$ where $N\geq 3$, $s\in[0,2)$, $p_s=\frac{2(N-s)}{N-2}$ and $\gamma\in (-\infty,\fra...

We perform a semiclassical analysis for the planar Schr\"odinger-Poisson system \[ \cases{ -\varepsilon^{2} \Delta\psi+V(x)\psi= E(x) \psi \quad \text{in $\mathbb{R}^2$},\cr -\Delta E= |\psi|^{2} \quad \text{in $\mathbb{R}^2$}, \cr } \tag{$SP_\varepsilon$} \] where $\varepsilon$ is a positive parameter corresponding to the Planck constant and $V$ i...

A partially hinged, partially free rectangular plate is considered, with the aim to address the possible unstable end behaviors of a suspension bridge subject to wind. This leads to a nonlinear plate evolution equation with a nonlocal stretching active in the span-wise direction. The wind-flow in the chord-wise direction is modeled through a piston...

This paper is devoted to the symmetry and symmetry breaking properties of a two-dimensional magnetic Schrödinger operator involving an Aharonov–Bohm magnetic vector potential. We investigate the symmetry properties of the optimal potential for the corresponding magnetic Keller–Lieb–Thirring inequality. We prove that this potential is radially symme...

In this paper, we consider the electrostatic Born-Infeld model(BI){−div(∇ϕ1−|∇ϕ|2)=ρin RN,lim|x|→∞⁡ϕ(x)=0 where ρ is a charge distribution on the boundary of a bounded domain Ω⊂RN, with N⩾3. We are interested in its equilibrium measures, i.e. charge distributions which minimize the electrostatic energy of the corresponding potential among all possi...

Fluid flows around an obstacle generate vortices which, in turn, generate lift forces on the obstacle. Therefore, even in a perfectly symmetric framework equilibrium positions may be asymmetric. We show that this is not the case for a Poiseuille flow in an unbounded 2D channel, at least for small Reynolds number and flow rate. We consider both the...

We consider nonlinear 2nd-order elliptic problems of the type $$\begin{align*} & -\Delta u=f(u)\ \textrm{in}\ \Omega, \qquad u=0\ \textrm{on}\ \partial \Omega, \end{align*}$$where $\Omega $ is an open $C^{1,1}$–domain in ${{\mathbb{R}}}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power...

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≥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...

We consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely, we prove that the best constant is achieved by a nonconstant solution of the associated fourth order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates...

Turbulence is a long-standing mystery. We survey some of the existing (and sometimes contradictory) results and suggest eight natural questions whose answers would increase the mathematical understanding of this phenomenon; each of these questions, yet, gives rise to ten subquestions.

In this note we prove the instability by blow-up of the ground state solutions for a class of fourth order Schrödinger equations. This extends the first rigorous results on blowing-up solutions for the biharmonic NLS due to Boulenger and Lenzmann [9] and confirm numerical conjectures from [1, 2, 3, 11].

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schr\"odinger equation $$ \gamma \Delta ^2 u -\Delta u + \alpha u=|u|^{2 \sigma} u, \quad u \in H^2(\R^N), $$ under the constraint $$ \int_{\R^N}|u|^2 \, dx =c>0. $$ We assume $\gamma >0, N \geq 1, 4 \leq \sigma N < \frac{4N}{(N-4)^+}$, whereas the parameter $\alph...

Turbulence is a long-standing mystery. We survey some of the existing (and sometimes contradictory) results and suggest eight natural questions whose answers would increase the mathematical understanding of this phenomenon; each of these questions, yet, gives rise to ten sub-questions.

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}\rho u\, dx, \end{equation*} where $\rho \in L^{m}(\mathbb{R}^N)$ is an assigned charge density, $m \in [1,2_*]$, $2_*:=\frac{2N}{N+2}$, $N\geq 3$. We prove that if $\rho \in L^q(\mathbb{R}^N) $ for $...

We consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain treshold, namely we prove that the best constant is achieved by a non-constant solution of the associated fourth-order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates...

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schr{\"o}dinger operators involving Aharonov-Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensiona...

This paper is devoted to the symmetry and symmetry breaking properties of a two-dimensional magnetic Schrödinger operator involving an Aharonov-Bohm magnetic vector potential. We investigate the symmetry properties of the optimal potential for the corresponding magnetic Keller-Lieb-Thir-ring inequality. We prove that this potential is radially symm...

In this paper, we consider the electrostatic Born-Infeld model \begin{equation*} \tag{$\mathcal{BI}$} \left\{ \begin{array}{rcll} -\operatorname{div}\left(\displaystyle\frac{\nabla \phi}{\sqrt{1-|\nabla \phi|^2}}\right)&=& \rho & \hbox{in }\mathbb{R}^N, \\[6mm] \displaystyle\lim_{|x|\to \infty}\phi(x)&=& 0 \end{array} \right. \end{equation*} where...

A thin and narrow rectangular plate having the two short edges hinged and the two long edges free is considered. A nonlinear nonlocal evolution equation describing the deformation of the plate is introduced: well-posedness and existence of periodic solutions are proved. The natural phase space is a particular second order Sobolev space that can be...

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 f...

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}\rho u\, dx, \end{equation*} where $\rho \in L^{m}(\mathbb{R}^N)$ is an assigned charge density, $m \in [1,2_*]$, $2_*:=\frac{2N}{N+2}$, $N\geq 3$. We prove that if $\rho \in L^q(\mathbb{R}^N) $ for $...

In this paper, we study the mixed dispersion fourth‐order nonlinear Helmholtz equation Δ2u−βΔu+αu=Γ|u|p−2uinRN,for positive, bounded and ZN‐periodic functions Γ in the following three cases: (a)α<0,β∈Ror(b)α>0,β<−2αor(c)α=0,β<0.Using the dual method of Evéquoz and Weth, we find solutions to this equation and establish some of their qualitative prop...

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

In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz equation $$ \Delta ^2 u -\beta \Delta u + \alpha u= \Gamma|u|^{p-2} u \quad\text{in } \R^N, $$ for positive, bounded and $\Z^N$-periodic functions $\Gamma$ in the following three cases: \begin{equation*} (a)\;\; \alpha<0,\beta \in \R \qquad\text{or}\qquad (b)\;\; \alpha>0...

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schr\"odinger equation $$ \gamma \Delta ^2 u -\Delta u + \alpha u=|u|^{2 \sigma} u, \quad u \in H^2(\R^N), $$ under the constraint $$ \int_{\R^N}|u|^2 \, dx =c>0. $$ We assume $\gamma >0, N \geq 1, 4 \leq \sigma N < \frac{4N}{(N-4)^+}$, whereas the parameter $\alph...

We study the mixed dispersion fourth order nonlinear Schrödinger equation i∂ tψ γ Δ 2ψ +β Δ ψ +| ψ | 2σ ψ = 0 in ℝ× ℝN, where γ , σ > 0 and β ϵ ℝ. We focus on standing wave solutions, namely, solutions of the form ψ (x, t) = eiα tu(x) for some α ϵ ℝ. This ansatz yields the fourth order elliptic equation γ Δ 2u β Δ u + α u = | u| 2σ u. We consider t...

In this paper we investigate the existence and uniqueness of spacelike radial graphs of prescribed mean curvature in the Lorentz-Minkowski space $\mathbb{L}^{n+1}$, for $n\geq 2$, spanning a given boundary datum lying on the hyperbolic space $\mathbb{H}^n$.

In this paper we investigate the existence and uniqueness of spacelike radial graphs of prescribed mean curvature in the Lorentz-Minkowski space $\mathbb{L}^{n+1}$, for $n\geq 2$, spanning a given boundary datum lying on the hyperbolic space $\mathbb{H}^n$.

We study the mixed dispersion fourth order nonlinear Schr\"odinger equation \begin{equation*} %\tag{\protect{4NLS}}\label{4nls} i \partial_t \psi -\gamma \Delta^2 \psi +\beta \Delta \psi +|\psi|^{2\sigma} \psi =0\ \text{in}\ \R \times\R^N, \end{equation*} where $\gamma,\sigma>0$ and $\beta \in \R$. We focus on standing wave solutions, namely soluti...

We consider the boundary value problem $$ \left\{ \begin{array}{rcll} -\Delta u+ u -\lambda e^u&=&0,\ u>0 & \mathrm{in}\ B_1(0)\\ \partial_\nu u&=&0&\mathrm{on}\ \partial B_1(0), \end{array}\right. $$ whose solutions correspond to steady states of the Keller--Segel system for chemotaxis. Here $B_1(0)$ is the unit disk, $\nu$ the outer normal to $\p...

In this paper, we study the static Born-Infeld equation −div u 1 − ||u| 2 = n k=1 a k δx k in R N , lim |x|→∞ u(x) = 0, where N ≥ 3, a k ∈ R for all k = 1,. .. , n, x k ∈ R N are the positions of the point charges, possibly non symmetrically distributed, and δx k is the Dirac delta distribution centered at x k. For this problem, we give explicit qu...

In this paper, we study the static Born-Infeld equation $$ -\mathrm{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)=\sum_{k=1}^n a_k\delta_{x_k}\quad\mbox{in }\mathbb R^N,\qquad \lim_{|x|\to\infty}u(x)=0, $$ 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 symme...

We consider the stationary Keller-Segel equation \begin{equation*} \begin{cases} -\Delta v+v=\lambda e^v, \quad v>0 \quad & \text{in }\Omega,\\ \partial_\nu v=0 &\text{on } \partial \Omega, \end{cases} \end{equation*} where $\Omega$ is a ball. In the regime $\lambda\to 0$, we study the radial bifurcations and we construct radial solutions by a glui...

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}.

In this paper, we construct several families of radial solutions for the stationary Keller–Segel equation. The first one consists in solutions which concentrate simultaneously on the interior and on the exterior boundary of an annulus. The second family is made of solutions on the unit ball which concentrate on an interior sphere while the solution...

In this note we consider the action functional $$\begin{aligned} \int _{\mathbb {R}\times \omega } \left( 1 - \sqrt{1 - |\nabla u|^{2}} + W(u) \right) \mathrm {d}\bar{x} \end{aligned}$$where W is a double well potential and \(\omega \) is a bounded domain of \(\mathbb {R}^{N-1}\). We prove existence, one-dimensionality and uniqueness (up to transla...

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation $$ - \Delta u + a u = \frac{1}{2 \pi} \Bigl[\ln \frac{1}{|x|}* |u|^2 \Bigr] \ u \qquad \text{in $\mathbb{R}^2$} $$ and we establish its nondegeneracy. For the corresponding three-dimensional problem, the nondegeneracy property of...

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation $$ - \Delta u + a u = \frac{1}{2 \pi} \Bigl[\ln \frac{1}{|x|}* |u|^2 \Bigr] \ u \qquad \text{in $\mathbb{R}^2$} $$ and we establish its nondegeneracy. For the corresponding three-dimensional problem, the nondegeneracy property of...

In this paper, we construct several families of radial solutions for the stationary Keller-Segel equation. The first one consists in solutions which concentrate simultaneously on the interior and on the exterior boundary of an annulus. The second family is made of solutions on the unit ball which concentrate on an interior sphere while the solution...

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 p...

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 p...

We study the qualitative properties of groundstates of the time-independent magnetic semilinear Schr\"odinger 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 solutio...

We study the qualitative properties of groundstates of the time-independent magnetic semilinear Schr\"odinger 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 solutio...

In this paper, we study the semiclassical limit for the stationary magnetic nonlinear Schrödinger equation (i∇ + A(x)) 2 u + V (x)u = |u| p−2 u, x ∈ R 3 , (0.1) where p > 2, A is a vector potential associated to a given magnetic field B, i.e ∇ × A = B and V is a nonnegative, scalar (electric) potential which can be singular at the origin and vanish...

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 plan...

We study the mixed dispersion fourth order nonlinear Schrödinger equation i∂tψ − γ∆ 2 ψ + β∆ψ + |ψ| 2σ ψ = 0 in R × R N , where γ, σ > 0 and β ∈ R. We focus on standing wave solutions, namely solutions of the form ψ(x, t) = e iαt u(x), for some α ∈ R. This ansatz yields the fourth-order elliptic equation γ∆ 2 u − β∆u + αu = |u| 2σ u. We consider tw...

In this note we consider the action functional \[ \int_{\mathbb{R} \times \omega} \left( 1 - \sqrt{ 1 - |\nabla u|^2 } + W(u) \right) \, \mathrm{d}t, \] where $W$ is a double well potential and $\omega$ is a bounded domain of $\mathbb{R}^{N-1}$. We prove existence, one-dimensionality and uniqueness (up to translation) of a smooth minimizing phase t...

We consider the stationary Keller-Segel equation \begin{equation*} \begin{cases} -\Delta v+v=\lambda e^v, \quad v>0 \quad & \text{in }\Omega,\\ \partial_\nu v=0 &\text{on } \partial \Omega, \end{cases} \end{equation*} where $\Omega$ is a ball. In the regime $\lambda\to 0$, we study the radial bifurcations and we construct radial solutions by a glui...

Assuming B(R) is a ball in ℝᴺ, we analyze the positive solutions of the problem -Δu = |u|ᵖ⁻²u in B(R), ∂ᵣu = 0 on ∂B(R), that branch out from the constant solution u=1 as p grows from 2 to +∞. 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 →...

Assuming $B_{R}$ is a ball in $\mathbb R^{N}$, we analyze the positive solutions of the problem \[ \begin{cases} -\Delta u+u= |u|^{p-2}u, &\text{ in } B_{R},\newline \partial_{\nu}u=0,&\text{ on } \partial B_{R}, \end{cases} \] that branch out from the constant solution $u=1$ as $p$ grows from $2$ to $+\infty$. The non-zero constant positive soluti...

We consider the stationary Keller-Segel equation −∆v + v = λe v , v > 0 in Ω, ∂ν v = 0 on ∂Ω, where Ω is a ball. In the regime λ → 0, we study the radial bifurcations and we construct radial solutions by a gluing variational method. For any given n ∈ N 0 , we build a solution having multiple layers at r 1 ,. .. , rn by which we mean that the soluti...

In this paper, we study the behavior as \(p\rightarrow \infty \) of eigenvalues and eigenfunctions of a system of p-Laplacians, that is $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _p u = \lambda \alpha u^{\alpha -1} v^\beta &{}\quad \Omega , \\ -\Delta _p v = \lambda \beta u^{\alpha } v^{\beta -1} &{}\quad \Omega , \\ u=v=0, &{} \quad \part...

In this paper we study the Neumann problem\begin{equation*}\begin{cases}-\Delta u+u=u^p \& \text{ in }B\_1 \\u \textgreater{} 0, \& \text{ in }B\_1 \\\partial\_\nu u=0 \& \text{ on } \partial B\_1,\end{cases}\end{equation*}and we show the existence of multiple-layer radial solutions as $p\rightarrow+\infty$.

In this note we provide some simple results for the 4NLS model i∂tψ + ∆ψ + |ψ| 2σ ψ − γ∆ 2 ψ = 0, where γ > 0. Our aim is to partially complete the discussion on waveguide solutions in [11, Section 4.1]. In particular, we show that in the model case with a Kerr nonlinearity (σ=1), the least energy waveguide solution ψ(t, x) = exp(iαt)u(x) with α >...

Consider a Hamiltonian system of type \[ -\Delta u=H_{v}(u,v),\ -\Delta v=H_{u}(u,v) \ \ \text{ in } \Omega, \qquad u,v=0 \text{ on } \partial \Omega \] where $H$ is a power-type nonlinearity, for instance $H(u,v)= |u|^p/p+|v|^q/q$, having subcritical growth, and $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N\geq 1$. The aim of this paper is to...

In this paper we study the behavior as p → ∞ of solutions up,q to −∆pu−∆qu = 0 in a bounded smooth domain Ω with a Lipschitz Dirichlet boundary datum u = g on ∂Ω. We find that there is a uniform limit of a subsequence of solutions, that is, there is pj → ∞ such that up j ,q → u∞ uniformly in Ω and we prove that this limit u∞ is a solution to a vari...

We prove existence and multiplicity results for sign-changing solutions of a prescribed mean curvature equation with Dirichlet boundary conditions. Our arguments involve a perturbation of the degenerate part , which allows us to use classical variational techniques and to localize small regular solutions.

We prove the existence of heteroclinic solutions of the prescribed curvature equation (eqution presented) where V is a double-well potential and a is asymptotic to a positive periodic function. Such an equation is meaningful in the modeling theory of reaction-diffiusion phenomena which feature saturation at large value of the gradient. According to...

We consider radial solutions of elliptic systems of the form{−Δu+u=a(|x|)f(u,v)in BR,−Δv+v=b(|x|)g(u,v)in BR,∂νu=∂νv=0on ∂BR, where essentially a, b are assumed to be radially nondecreasing weights and f, g are nondecreasing in each component. With few assumptions on the nonlinearities, we prove the existence of at least one couple of nondecreasing...

We consider the ground state solutions of the Lane–Emden system with Hénon-type weights −Δu=|x|β|v|q−1v−Δu=|x|β|v|q−1v, −Δv=|x|α|u|p−1u−Δv=|x|α|u|p−1u in the unit ball B of RNRN with Dirichlet boundary conditions, where N⩾1N⩾1, α,β⩾0α,β⩾0, p,q>0p,q>0 and 1/(p+1)+1/(q+1)>(N−2)/N1/(p+1)+1/(q+1)>(N−2)/N. We show that such ground state solutions u, v a...

We study the existence of heteroclinics connecting the two equi-libria ±1 of the third order differential equation u = f (u) + p(t)u 1 Partially supported by Ministerio de Educación y Ciencia, Spain, Project MTM2010-15314. 2 Supported by Fundação para a Ciência e a Tecnologia, Financiamento Base ISL-209 (2010) and PEst-OE/MAT/UI0209/2011 . 1 where...

In this paper, we show that the quasilinear equation has a positive smooth radial solution at least for any α > 2* = 2N/(N- 2), N ≥ 3. Our approach is based on the study of the optimizers for the best constant in the inequality, which holds true in the unit ball of W1,1(ℝN)\D1;2(ℝN) if and only if and α2*. We also prove that the best constant is no...

We study positive bound states for the equation -epsilon(2) Delta u + V(x)u = K(x)f(u), x is an element of R(N). where epsilon > 0 is a real parameter and V and K are radial positive potentials. We are especially interested in solutions which concentrate on a k-dimensional sphere, 1 <= k <= N - 1, as epsilon -> 0. We adopt a purely variational appr...

The present paper is devoted to weighted Nonlinear Schr\"odinger- Poisson systems with potentials possibly unbounded and vanishing at infinity. Using a purely variational approach, we prove the existence of solutions concentrating on a circle.

We study the existence of positive increasing radial solutions for superlinear Neumann problems in the ball. We do not impose any growth condition on the nonlinearity at infinity and our assumptions allow for interactions with the spectrum. In our approach we use both topological and variational arguments, and we overcome the lack of compactness by...

We study the existence of positive solutions for the model scalar second-order boundary value problem where a, b, c are locally bounded coefficients and p>0.

We study an elliptic system of the form Lu = vertical bar v vertical bar(p-1) v and Lv = vertical bar u vertical bar(q-1) u in Omega with homogeneous Dirichlet boundary condition, where Lu := -Delta u in the case of a bounded domain and Lu := -Delta u + u in the cases of an exterior domain or the whole space R-N. We analyze the existence, uniquenes...

Motivated by existence results for positive solutions of non-autonomous nonlinear Schrödinger–Poisson systems with potentials possibly unbounded or vanishing at infinity, we prove embedding theorems for weighted Sobolev spaces. We both consider a general framework and spaces of radially symmetric functions when assuming radial symmetry of the poten...

We present some topological isomorphisms between Wm,p((0,1)) and Lp((0,1))×Rm. Combined with some arguments from Fourier analysis, we obtain explicit Schauder bases for Wm,p((0,1)) and some of its subspaces. In addition, we apply such results to treat some boundary value problems involving nonlinear elliptic differential equations.

We study positive bound states for the semiclassical stationary nonlinear Schr\"odinger equation. We are especially interested in solutions which concentrate on a lower dimensional sphere. We adopt a purely variational approach which allows us to consider broader classes of potentials than those treated in previous works. For example, the potential...

We consider a classical semilinear elliptic equation with Neumann boundary conditions on an annulus in R N . The nonlinear term is the product of a radially symmetric coefficient with a pure power. We prove that if the power is sufficiently large, the problem admits at least three distinct positive and radial solutions. In case the coefficient is...

Groundstates of the stationary nonlinear Schrödinger equation $$-\Delta u +V u =K u^{p-1}$$, are studied when the nonnegative function V and K are neither bounded away from zero, nor bounded from above. A special attention is paid in the case of a potential V that goes to 0 at infinity. Conditions on compact embeddings that allow to prove in partic...