arXiv subjects

Louis Jeanjean

Publications and source records attributed to Louis Jeanjean.

At least 37 records · Page 2Linked to original sources

arXiv2021

Logarithmic estimates for mean-field models in dimension two and the Schrödinger-Poisson system

In dimension two, we investigate a free energy and the ground state energy of the Schrödinger-Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling invariances of the problem. Such a system can be considered as a nonlinear Schrödinger equation with a cubic but nonlocal Poisson nonlinearity, and a local logarithmic nonlinearity. Both cases of repulsive and attractive forces are considered. We also assume that there is an external potential with minimal growth at infinity, which turns out to have a logarithmic growth. Our estimates rely on new logarithmic interpolation inequalities which combine logarithmic Hardy-Littlewood-Sobolev and logarithmic Sobolev inequalities. The two-dimensional model appears as a limit case of more classical problems in higher dimensions.

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arXiv2021

Multiple normalized solutions for a Sobolev critical Schrödinger equation

We study the existence of standing waves, of prescribed $L^2$-norm (the mass), for the nonlinear Schrödinger equation with mixed power nonlinearities $$ i \partial_t ϕ+ Δϕ+ μϕ|ϕ|^{q-2} + ϕ|ϕ|^{2^* - 2} = 0, \quad (t, x) \in R \times R^N, $$ where $N \geq 3$, $ϕ: R \times R^N \to C$, $μ> 0$, $2 < q < 2 + 4/N $ and $2^* = 2N/(N-2)$ is the critical Sobolev exponent. It was already proved that, for small mass, ground states exist and correspond to local minima of the associated Energy functional. It was also established that despite the nonlinearity is Sobolev critical, the set of ground states is orbitally stable. Here we prove that, when $N \geq 4$, there also exist standing waves which are not ground states and are located at a mountain-pass level of the Energy functional. These solutions are unstable by blow-up in finite time. Our study is motivated by a question raised by N. Soave.

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arXiv2020

Some remarks on a minimization problem associated to a fourth order nonlinear Schrödinger equation

Let $γ> 0\,$, $β> 0\,$, $α> 0$ and $0 < σN < 4$. In the present paper, we study, for $c > 0$ given, the constrained minimization problem \begin{equation*} \label{MinL2fixed} m(c):=\inf_{u\in S (c) }E(u), \end{equation*} where \begin{equation*} E (u):=\fracγ{2}\int_{\mathbb{R}^N}|Δu|^2\, dx -\fracβ{2}\int_{\mathbb{R}^N}|\nabla u|^2\, dx-\fracα{2σ+2}\int_{\mathbb{R}^N}|u|^{2σ+2}\, dx, \end{equation*} and \begin{equation*} S(c):=\left\{u\in H^2(\mathbb{R}^N):\int_{\mathbb{R}^N}|u|^{2}\, dx=c\right\}. \end{equation*} The aim of our study is twofold. On one hand, this minimization problem is related to the existence and orbital stability of standing waves for the mixed dispersion nonlinear biharmonic Schrödinger equation \begin{equation*} i \partial_t ψ-γΔ^2 ψ- βΔψ+ α|ψ|^{2σ} ψ=0, \quad ψ(0, x)=ψ_0 (x),\quad (t, x) \in \mathbb{R} \times \mathbb{R}^N. \end{equation*} On the other hand, in most of the applications of the Concentration-Compactness principle of P.-L. Lions, the difficult part is to deal with the possible dichotomy of the minimizing sequences. The problem under consideration provides an example for which, to rule out the dichotomy is rather standard while, to rule out the vanishing, here for $c > 0$ small, is challenging. We also provide, in the limit $c \to 0$, a precise description of the behaviour of the minima. Finally, some extensions and open problems are proposed.

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arXiv2020

A mass supercritical problem revisited

In any dimension $N\geq1$ and for given mass $m>0$, we revisit the nonlinear scalar field equation with an $L^2$ constraint: $$ -Δu=f(u)-μu, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ where $μ\in\mathbb{R}$ will arise as a Lagrange multiplier. Assuming only that the nonlinearity $f$ is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states and reveal the basic behavior of the ground state energy $E_m$ as $m>0$ varies. In particular, to overcome the compactness issue when looking for ground states, we develop robust arguments which we believe will allow treating other $L^2$ constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any $N\geq2$ and establish the existence and multiplicity of nonradial sign-changing solutions when $N\geq4$. Finally we propose two open problems.

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arXiv2019

Nonradial normalized solutions for nonlinear scalar field equations

We study the following nonlinear scalar field equation $$ -Δu=f(u)-μu, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ Here $f\in C(\mathbb{R},\mathbb{R})$, $m>0$ is a given constant and $μ\in\mathbb{R}$ is a Lagrange multiplier. In a mass subcritical case but under general assumptions on the nonlinearity $f$, we show the existence of one nonradial solution for any $N\geq4$, and obtain multiple (sometimes infinitely many) nonradial solutions when $N=4$ or $N\geq6$. In particular, all these solutions are sign-changing.

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arXiv2019

Stationary waves with prescribed $L^2$-norm for the planar Schrödinger-Poisson system

The paper deals with the existence of standing wave solutions for the Schrödinger-Poisson system with prescribed mass in dimension $N=2$. This leads to investigate the existence of normalized solutions for an integro-differential equation involving a logarithmic convolution potential, namely $$ \left \{ \begin{aligned} - Δu & + λu + γ\Bigl(\log {| \cdot |} * |u|^2 \Bigr) u =a |u|^{p-2} u \qquad \text{in $\mathbb R^2$,} \\ &\int_{\mathbb R^2} |u|^2 dx = c \end{aligned} \right. $$ where $c>0$ is a given real number. Under different assumptions on $γ\in \mathbb R$, $a \in \mathbb R$, $p>2$, we prove several existence and multiplicity results. Here $λ\in \mathbb R $ appears as a Lagrange parameter and is part of the unknowns. With respect to the related higher dimensional cases, the presence of the logarithmic kernel, which is unbounded from above and below, makes the structure of the solution set much richer and it forces the implementation of new ideas to catch the normalized solutions.

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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

A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient

Let $Ω\subset \mathbb R^N$, $N \geq 2$, be a smooth bounded domain. We consider a boundary value problem of the form $$-Δu = c_λ(x) u + μ(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(Ω)\cap L^{\infty}(Ω)$$ where $c_λ$ depends on a parameter $λ\in \mathbb R$, the coefficients $c_λ$ and $h$ belong to $L^q(Ω)$ with $q>N/2$ and $μ\in L^{\infty}(Ω)$. Under suitable assumptions, but without imposing a sign condition on any of these coefficients, we obtain an a priori upper bound on the solutions. Our proof relies on a new boundary weak Harnack inequality. This inequality, which is of independent interest, is established in the general framework of the $p$-Laplacian. With this a priori bound at hand, we show the existence and multiplicity of solutions.

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arXiv2018

Nonlinear scalar field equations with general nonlinearity

Consider the nonlinear scalar field equation \begin{equation} \label{a1} -Δ{u}= f(u)\quad\text{in}~\mathbb{R}^N,\qquad u\in H^1(\mathbb{R}^N), \end{equation} where $N\geq3$ and $f$ satisfies the general Berestycki-Lions conditions. We are interested in the existence of positive ground states, of nonradial solutions and in the multiplicity of radial and nonradial solutions. Very recently Mederski [30] made a major advance in that direction through the development, in an abstract setting, of a new critical point theory for constrained functionals. In this paper we propose an alternative, more elementary approach, which permits to recover Mederski's results on the scalar field equation. The keys to our approach are an extension to the symmetric mountain pass setting of the monotonicity trick, and a new decomposition result for bounded Palais-Smale sequences.

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arXiv2018

Multiple positive normalized solutions for nonlinear Schrödinger systems

We consider the existence of multiple positive solutions to the nonlinear Schrödinger systems sets on $H^1(\mathbb{R}^N) \times H^1(\mathbb{R}^N)$, \[ \left\{ \begin{aligned} -Δu_1 &= λ_1 u_1 + μ_1 |u_1|^{p_1 -2}u_1 + βr_1 |u_1|^{r_1-2} u_1|u_2|^{r_2}, -Δu_2 &= λ_2 u_2 + μ_2 |u_2|^{p_2 -2}u_2 + βr_2 |u_1|^{r_1} |u_2|^{r_2 -2} u_2, \end{aligned} \right. \] under the constraint \[ \int_{\mathbb{R}^N}|u_1|^2 \, dx = a_1,\quad \int_{\mathbb{R}^N}|u_2|^2 \, dx = a_2. \] Here $a_1, a_2 >0$ are prescribed, $μ_1, μ_2, β>0$, and the frequencies $λ_1, λ_2$ are unknown and will appear as Lagrange multipliers. Two cases are studied, the first when $N \geq 1, 2 < p_1, p_2 < 2 + \frac 4N, r_1, r_2 > 1, 2 + \frac 4N < r_1 + r_2 < 2^*$, the second when $ N \geq 1, 2+ \frac 4N < p_1, p_2 < 2^*, r_1, r_2 > 1, r_1 + r_2 < 2 + \frac 4N.$ In both cases, assuming that $β>0$ is sufficiently small, we prove the existence of two positive solutions. The first one is a local minimizer for which we establish the compactness of the minimizing sequences and also discuss the orbital stability of the associated standing waves. The second solution is obtained through a constrained mountain pass and a constrained linking respectively.

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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

Multiple complex-valued solutions for nonlinear magnetic Schrodinger equations

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations $$ L^{\hbar}_{A,V} u = f(|u|^2)u \quad \mbox{in } R^N $$ where $N \geq 3$, $L^{\hbar}_{A,V}$ is the Schrödinger operator with a magnetic field having source in a $C^1$ vector potential $A$ and a scalar continuous (electric) potential $V$ defined by \begin{equation} L^{\hbar}_{A,V}= -\hbar^2 Δ-\frac{2\hbar}{i} A \cdot \nabla + |A|^2- \frac{\hbar}{i}\operatorname{div}A + V(x). \end{equation} Here $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain $Ω\subset R^N$ such that \[ m_0 \equiv \inf_{x \in Ω} V(x) < \inf_{x \in \partial Ω} V(x) \] and we set $K = \{ x \in Ω\ | \ V(x) = m_0\}$. For $\hbar >0$ small we prove the existence of at least $cuplenght(K) + 1$ geometrically distinct, complex-valued solutions whose modula concentrate around $K$ as $\hbar \to 0$.

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arXiv2016

On dipolar quantum gases in the unstable regime

We study the nonlinear Schrödinger equation arising in dipolar Bose-Einstein condensate in the unstable regime. Two cases are studied: the first when the system is free, the second when gradually a trapping potential is added. In both cases we first focus on the existence and stability/ instability properties of standing waves. Our approach leads to the search of critical points of a constrained functional which is unbounded from below on the constraint. In the free case, by showing that the constrained functional has a so-called {\it mountain pass geometry}, we prove the existence of standing states with least energy, the ground states, and show that any ground state is orbitally unstable. Moreover, when the system is free, we show that small data in the energy space scatter in all regimes, stable and unstable. In the second case, if the trapping potential is small, we prove that two different kind of standing waves appears: one corresponds to a {\it topological local minimizer} of the constrained energy functional and it consists in ground states, the other is again of {\it mountain pass type} but now corresponds to excited states. We also prove that any ground state is a {\it topological local minimizer}. Despite the problem is mass supercritical and the functional unbounded from below, the standing waves associated to the set of ground states turn to be orbitally stable. Actually, from the physical point of view, the introduction of the trapping potential stabilizes the system initially unstable. Related to this we observe that it also creates a gap in the ground state energy level of the system. In addition when the trapping potential is active the presence of standing waves with arbitrary small norm does not permit small data scattering. Eventually some asymptotic results are also given.

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arXiv2016

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

In this paper we investigate the existence of solutions in $H^1(R^N) \times H^1(R^N)$ for nonlinear Schrödinger systems of the form \[ \left\{ \begin{aligned} -Δu_1 &= λ_1 u_1 + μ_1 |u_1|^{p_1 -2}u_1 + r_1β|u_1|^{r_1-2}u_1|u_2|^{r_2}, \\ -Δu_2 &= λ_2 u_2 + μ_2 |u_2|^{p_2 -2}u_2 + r_2 β|u_1|^{r_1}|u_2|^{r_2 -2}u_2, \end{aligned} \right. \] under the constraints \[\int_{R^N}|u_1|^2 \, dx = a_1>0,\quad \int_{R^N}|u_2|^2 \, dx = a_2>0. \] Here $ N \geq 1, β>0, μ_i >0, r_i >1, 2 <p_i < 2 + \frac{4}{N}$ for $i=1,2$ and $ r_1 + r_2 < 2 + \frac{4}{N}$. This problem is motivated by the search of standing waves for an evolution problem appearing in several physical models. Our solutions are obtained as constrained global minimizers of an associated functional. Note that in the system $λ_1$ and $λ_2$ are unknown and will correspond to the Lagrange multipliers. Our main result is the precompactness of the minimizing sequences, up to translation, and as a consequence we obtain the orbital stability of the standing waves associated to the set of minimizers.

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arXiv2015

Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient

We consider the boundary value problem \begin{equation} - Δu = λc(x)u+ μ(x) |\nabla u|^2 + h(x), \qquad u \in H^1_0(Ω) \cap L^{\infty}(Ω), \leqno{(P_λ)} \end{equation} where $Ω\subset \R^N, N \geq 3$ is a bounded domain with smooth boundary. It is assumed that $c\gneqq 0$, $c,h$ belong to $L^p(Ω)$ for some $p > N$. Also $μ\in L^{\infty}(Ω)$ and $μ\geq μ_1 >0$ for some $μ_1 \in \R$. It is known that when $λ\leq 0$, problem $(P_λ)$ has at most one solution. In this paper we study, under various assumptions, the structure of the set of solutions of $(P_λ)$ assuming that $λ>0$. Our study unveils the rich structure of this problem. We show, in particular, that what happen for $λ=0$ influences the set of solutions in all the half-space $]0,+\infty[\times(H^1_0(Ω) \cap L^{\infty}(Ω))$. Most of our results are valid without assuming that $h$ has a sign. If we require $h$ to have a sign, we observe that the set of solutions differs completely for $h\gneqq 0$ and $h\lneqq 0$. We also show when $h$ has a sign that solutions not having this sign may exists. Some uniqueness results of signed solutions are also derived. The paper ends with a list of open problems.

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arXiv2015

Normalized solutions for nonlinear Schrödinger systems

We consider the existence of \emph{normalized} solutions in $H^1(\R^N) \times H^1(\R^N)$ for systems of nonlinear Schrödinger equations which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz one is led to coupled systems of elliptic equations of the form \[ \left\{ \begin{aligned} -\De u_1 &= \la_1u_1 + f_1(u_1)+\pa_1F(u_1,u_2),\\ -\De u_2 &= \la_2u_2 + f_2(u_2)+\pa_2F(u_1,u_2),\\ u_1,u_2&\in H^1(\R^N),\ N\ge2, \end{aligned} \right. \] and we are looking for solutions satisfying \[ \int_{\R^N}|u_1|^2 = a_1,\quad \int_{\R^N}|u_2|^2 = a_2 \] where $a_1>0$ and $a_2>0$ are prescribed. In the system $\la_1$ and $\la_2$ are unknown and will appear as Lagrange multipliers. We treat the case of homogeneous nonlinearities, i.e.\ $f_i(u_i)=μ_i|u_i|^{p_i-1}u_i$, $F(u_1,u_2)=\be|u_1|^{r_1}|u_2|^{r_2}$, with positive constants $\be, μ_i, p_i, r_i$. The exponents are Sobolev subcritical but may be $L^2$-supercritical: $p_1,p_2,r_1+r_2\in]2,2^*[\,\setminus\left\{2+\frac4N\right\}$.

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arXiv2015

Normalized solutions for a system of coupled cubic Schrödinger equations on $\mathbb{R}^3$

We consider the system of coupled elliptic equations \[ \begin{cases} -Δu - λ_1 u = μ_1 u^3+ βu v^2 \\ -Δv- λ_2 v = μ_2 v^3 +βu^2 v \end{cases} \text{in $\mathbb{R}^3$}, \] and study the existence of positive solutions satisfying the additional condition \[ \int_{\mathbb{R}^3} u^2 = a_1^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a_2^2. \] Assuming that $a_1,a_2,μ_1,μ_2$ are positive fixed quantities, we prove existence results for different ranges of the coupling parameter $β>0$. The extension to systems with an arbitrary number of components is discussed, as well as the orbital stability of the corresponding standing waves for the related Schrödinger systems.

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arXiv2015

Multiple normalized solutions for quasi-linear Schrödinger equations

In this paper we prove the existence of two solutions having a prescribed $L^2$-norm for a quasi-linear Schrödinger equation. One of these solutions is a mountain pass solution relative to a constraint and the other one a minimum either local or global. To overcome the lack of differentiability of the associated functional, we rely on a perturbation method developed in [27].

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