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Hichem Hajaiej

Publications and source records attributed to Hichem Hajaiej.

At least 37 records · Page 2Linked to original sources
arXiv2023

Threshold for Blowup and Stability for Nonlinear Schrödinger Equation with Rotation

We consider the focusing NLS with an angular momentum and a harmonic potential, which models Bose-Einstein condensate under a rotating magnetic trap. We give a sharp condition on the global existence and blowup in the mass-critical case. We further consider the stability of such systems via variational method. We determine that at the critical exponent $p=1+4/n$, the mass of $Q$, the ground state for the NLS with zero potential, is the threshold for both finite time blowup and orbital instability. Moreover, we prove a sharp threshold theorem for the rotational NLS with an inhomogeneous nonlinearity. The analysis relies on the existence of ground state as well as a virial identity for the associated kinetic-magnetic operator.

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arXiv2023

Existence and non-existence results to a mixed Schrodinger system in a plane

This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type \begin{equation*} \left\{ \begin{aligned} -\partial_{xx}u + (-Δ)_{y}^{s_{1}} u + u - u^{2_{s_{1}}^{}-1} = καh(x,y) u^{α-1}v^β & \quad \mbox{in} ~ \mathbb{R}^{2}, -\partial_{xx}v + (-Δ)_{y}^{s_{2}} v + v- v^{2_{s_{2}}^{}-1} = κβh(x,y) u^αv^{β-1} & \quad \mbox{in} ~ \mathbb{R}^{2}, u,v ~ \geq ~0 \quad \mbox{in} ~ \mathbb{R}^{2}, \end{aligned} \right. \end{equation*} where $s_{1},s_{2} \in (0,1),~α,β>1,~α+β\leq \min \{ 2_{s_{1}}^{},2_{s_{2}}^{}\}$, and $2_{s_i}^{} = \frac{2(1+s_i)}{1-s_i}, i=1,2$. The existence of a ground state solution entirely depends on the behaviour of the parameter $κ>0$ and on the function $h$. In this article, we prove that a ground state solution exists in the subcritical case if $κ$ is large enough and $h$ satisfies (1.3). Further, if $κ$ becomes very small in this case then there does not exist any solution to our system. The study in the critical case, i.e. $s_1=s_2=s, α+β=2_s$, is more complex and the solution exists only for large $κ$ and radial $h$ satisfying (H1). Finally, we establish a Pohozaev identity which enables us to prove the non-existence results under some smooth assumptions on $h$.

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arXiv2023

Normalized Ground State Solutions for Fractional Schrodinger Systems with General Nonlinearities

Normalized ground state solutions (NGSS) of Schrodinger equations (SE) have attracted the attention of many research groups during the last decades. This is essentially due to their relevance in many fields in physics and engineering, where the stable and most attractive solutions happen to be the normalized ones. For a single (SE), recent developments lead to the establishment of existence and non-existence results for a wide range of natural nonlinearities in (SE) in the sub-critical, critical and super-critical regimes. However for systems of (SE), there are still many interesting open questions for basic nonlinearities. It certainly requires innovative ideas to shed some light to treat these complex situations. So far, only a very few specific nonlinearities have been addressed. Unlike the single (SE), the corresponding strict sub-additivity inequality is challenging and an improved concentration-compactness theorem is critical to treat (NGSS) of systems of (SE). The aim of this paper is to establish the existence of (NGSS) for a large class of nonlinearities. This class includes many relevant pure-power type nonlinearities and can be easily extended to Hartree type nonlinearities (and a combination of both). The presence of the fractional Laplacian adds a considerable difficulty to rule out the dichotomy. We were able to overcome this challenge and establish very general assumptions ensuring the strict subadditivity of the constrained energy functional. We believe that our approach will open the door to many unresolved problems.

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arXiv2023

On the Number of Normalized Ground State Solutions for a class of Elliptic Equations with general nonlinearities and potentials

We provide a precise description of the set of normalized ground state solutions (NGSS) for the class of elliptic equations: $$ -Δu - λu + V (| x |) u - f (| x |, u) = 0,\quad\text{in}\quad \mathbb{R}^n,\ n\geq 1. $$ In particular, we show that under suitable assumptions on $V$ and $f$, the NGSS is unique for all the masses except for at most a finite number. Moreover, we prove that when unique, the NGSS $u_c$ is a smooth function of the mass $c.$ Our method is as follow: using the NGSS for a given mass $c$, we construct an exhaustive list of potential candidates to the minimization problem for masses close to $c$, and we develop a strategy how to pick the right one. In particular, if there is a unique NGSS for a given mass $c_0,$ then this uniqueness property is inherited for all the masses $c$ close to $c_0.$ Our method is general and applies to other equations provided that some key properties hold true.

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arXiv2023

Existence and Multiplicity of Normalized Solutions for Dirac Equations with non-autonomous nonlinearities

In this paper, we study the following nonlinear Dirac equations \begin{align*} \begin{cases} -i\sum\limits_{k=1}^3α_k\partial_k u+mβu=f(x,|u|)u+ωu, \displaystyle \int_{\mathbb{R}^3} |u|^2dx=a^2, \end{cases} \end{align*} where $u: \mathbb{R}^{3}\rightarrow \mathbb{C}^{4}$, $m>0$ is the mass of the Dirac particle, $ω\in \mathbb{R}$ arises as a Lagrange multiplier, $\partial_k=\frac{\partial}{\partial x_k}$, $α_1,α_2,α_3$ are $4\times 4$ Pauli-Dirac matrices, $a>0$ is a prescribed constant, and $f(x,\cdot)$ has several physical interpretations that will be discussed in the Introduction. Under general assumptions on the nonlinearity $f$, we prove the existence of $L^2$-normalized solutions for the above nonlinear Dirac equations by using perturbation methods in combination with Lyapunov-Schmidt reduction. We also show the multiplicity of these normalized solutions thanks to the multiplicity theorem of Ljusternik-Schnirelmann. Moreover, we obtain bifurcation results of this problem.

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arXiv2023

On the focusing fractional nonlinear Schrödinger equation on the waveguide manifolds

In this paper, we consider the focusing fractional nonlinear Schrödinger equation (FNLS) on the waveguide manifolds $\mathbb{R}^d\times\mathbb{T}^m$ both in the isotropic and anisotropic case. Under different conditions, we establish the existence and periodic dependence of the ground states of the focusing FNLS. In the intercritical regime, we also establish the large data scattering for the anisotropic focusing FNLS by appealing to the framework of semivirial vanishing geometry.

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arXiv2023

Existence and instability of standing waves for the biharmonic nonlinear Schroedinger equation with combined nonlinearities

We prove the existence of normalized ground state solutions for the biharmonic Schrödinger equation with combined nonlinearities and show that all ground states correspond to the local minima of the associated energy functional restricted to the appropriate set. Moreover, we prove that the standing waves are strongly unstable by blowup. In particular, our results cover the critical case. Our method is novel and innovative as previous ideas cannot be used in many cases under this study.

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arXiv2023

Positive solution for an elliptic system with critical exponent and logarithmic terms

In this paper, we study the existence and nonexistence of positive solutions for a coupled elliptic system with critical exponent and logarithmic terms. The presence of the the logarithmic terms brings major challenges and makes it difficult to use the previous results established in the work of Chen and Zou without new ideas and innovative techniques.

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arXiv2023

A General and Unified Method to prove the Uniqueness of Ground State Solutions and the Existence/Non-existence, and Multiplicity of Normalized Solutions with applications to various NLS

We first give an abstract framework to show the uniqueness of Ground State Solutions (GSS) for a large class of PDEs. To the best of our knowledge, all the existing results in the literature only addressed particular cases. Moreover, our self-contained approach offers a general framework to study the existence/non-existence and multiplicity of normalized solutions. We will exhibit concrete examples to which our method applies, and verify all the assumptions we need. Our approach is applicable to a wide range of operators and domains provided that our hypotheses are verified. Additionally, we prove new results about the non-degeneracy and uniqueness of positive GSS in a general setting. Our findings are applicable to fractional nonlinear Schrodinger equations with non-autonomous nonlinearities. In particular, we were able to extend the main results of [13, 14] to general non-autonomous and mixed nonlinearities. This does not seem possible by using the approach developed by the authors of the above breakthrough papers. The orbital stability/instability of the standing waves will be addressed thanks to the non-degeneracy.

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arXiv2023

Standing waves for the NLS equation with competing nonlocal and local nonlinearities: the double $L^{2}$-supercritical case

We investigate the NLS equation with competing Hartree-type and power-type nonlinearities \begin{equation*} \begin{array}{ll} i\partial _{t}ψ+Δψ+γ(I_{α}\ast |ψ|^{p})|ψ|^{p-2}ψ+μ|ψ|^{q-2}ψ=0, & \text{ }\forall (t,x)\in \mathbb{R\times R}^{N},% \end{array}% \end{equation*}% where $γμ<0$. We establish conditions for the local well-posedness in the energy space. Under the double $L^{2}$-supercritical case, we prove the existence and multiplicity of standing waves with prescribed mass by developing a constraint method when $γ<0,μ>0$ and $γ>0,μ<0, $ respectively. Moreover, we prove weak orbital stablility and strong instability of standing waves by considering a suitable local minimization problem and by analyzing the fibering mapping, respectively. A new analysis of the fibering mapping is performed in this work. We believe that it is innovative as it was not discussed at all in any previous results. The lower bound rate of blow-up solutions for the Cauchy problem is given as well. Due to the different \textquotedblleft strength" of the two types of nonlinearities, we find some essential differences in our results between these two competing cases. We will be dealing with two major scenarios that are totally different from each other due to their diverse geometric structure. This leads to surprising findings. Additionally, the competing pure power-type nonlinearities case can be derived from our study thanks to a good choice of the kernel of the Hartree term.

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arXiv2023

Liouville Theorem for semilinear elliptic inequalities with the fractional Hardy operators

In this paper, we give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains. Our methods are self-contained and new. The main ideas and key ingredients will be discussed in the next section after Theorem 1.1 for punctured domains, and after Theorem 1.4 for exterior domains. We will also explain why all the previous methods and techniques do not apply to our general setting. Let us give here a foretaste of our line of attack: Based on the imbalance between the Hardy operator and the nonlinearity, we can obtain an initial asymptotic behavior rate at the origin ( for punctured domains) or at infinity ( exterior domains).We then improve this rate by using the interaction with the nonlinearity. By repeating this process finite number of times, a contradiction will be deduced from the nonexistence for the related non-homogeneous fractional Hardy problem. This process allows us to obtain the nonexistence for the fractional Hardy problem with larger ranges . Our study covers all possible ranges, and our results are optimal.

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arXiv2023

On the solitary waves for anisotropic nonlinear Schrödinger models on the plane

The focussing anisotropic nonlinear Schrödinger equation \begin{align*} \mathrm{i} u_t-\partial_{xx} u + (-\partial_{yy})^s u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R} \times \mathbb{R}^2 \end{align*} is considered for $0 2$. Here the equation is of anisotropy, it means that dispersion of solutions along $x$-axis and $y$-axis is different. We show that while localized time-periodic waves, that are solutions in the form $u=e^{-\mathrm{i} ωt} ϕ$, do not exist in the regime $p\geq p_s:=\frac{2(1+s)}{1-s}$, they do exist in the complementary regime $2<p<p_s$. In fact, we construct them variationally and we establish a number of key properties. Importantly, we completely characterize their spectral stability properties. Our consideration are easily extendable to the higher dimensional situation. We also show uniqueness of these waves under a natural weak non-degeneracy assumption. This assumption is actually removed for $s$ close to $1$, implying uniqueness for the waves in the full range of parameters.

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arXiv2023

On the Eigenvalues of the $p\&q-$ Fractional Laplacian

We consider the eigenvalue problem for the fractional $p \& q-$Laplacian \begin{equation} \left\{\begin{aligned} (- Δ)_p^{s}\, u + μ(- Δ)_q^{s}\, u+ |u|^{p-2}u+μ|u|^{q-2}u=λ V(x)|u|^{p-2}u\quad & \text{in } Ω\\ u=0\quad& \text{in}\quad\R^N\backslashΩ, \end{aligned}\right. \end{equation} where $Ω$ is an open bounded, and possibly disconnected domain, $λ\in\R$, $1 0$ with a weight function in $L^\infty(Ω)$ that is allowed no change sign. We show that the problem has a continuous spectrum. Moreover, our result reveals a discontinuity property for the spectrum as the parameter $μ\to 0^+.$ In addition, a stability property of eigenvalues as $s\to 1^-$ is established.

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arXiv2022

On existence and stability results for normalized ground states of mass-subcritical biharmonic NLS on $\mathbb{R}^d\times\mathbb{T}^n$

We study the focusing mass-subcritical biharmonic nonlinear Schrödinger equation (BNLS) on the product space $\mathbb{R}_x^d\times\mathbb{T}_y^n$. Following the crucial scaling arguments introduced in \cite{TTVproduct2014} we establish existence and stability results for the normalized ground states of BNLS. Moreover, in the case where lower order dispersion is absent, we prove the existence of a critical mass number $c_0\in(0,\infty)$ that sharply determines the $y$-dependence of the deduced ground states. In the mixed dispersion case, we encounter a major challenge as the BNLS is no longer scale-invariant and the arguments from \cite{TTVproduct2014} for determining the sharp $y$-dependence of the ground states fail. The main novelty of the present paper is to address this difficult and interesting issue: Using a different scaling argument, we show that $y$-independence of ground states with small mass still holds in the case $β>0$ and $α\in(0,4/(d+n))$. Additionally, we also prove that ground states with sufficiently large mass must possess non-trivial $y$-dependence by appealing to some novel construction of test functions. The latter particularly holds for all parameters lying in the full mass-subcritical regime.

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arXiv2022

A New Method to prove the Existence, Non-existence, Multiplicity, Uniqueness, and Orbital Stability/Instability of standing waves for NLS with partial confinement

We give a new method to prove the existence, non-existence, multiplicity, orbital stability/instability of standing waves for NLS with partial confinement without the subcritical hypothesis, even in the reduction equation. Using this method, we give an affirmative answer for an open problem proposed by [7, Remark 1.10] where the authors conjectured the existence of more than a normalized solution. We also establish uniqueness results of the ground state solutions depending on the bifurcation parameters. We explain that when the effect of partial confinement is strong, a dimension reduction appears for some parameters. We also find different bifurcation phenomena from the cases with full confinement.

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arXiv2022

On isolated singularities for fractional Lane-Emden equation in the Serrin's critical case

In this paper, we solve the fractional Lane-Emden equation in the Serrin's critical case for the fractional Laplacian by developing an innovative and self-contained approach that also applies to the classical setting ( Laplacian). We give a classification of the isolated singularities of positive solutions to the semilinear fractional elliptic equations $$(E) \qquad\qquad (-Δ)^s u = u^{\frac{N}{N-2s}}\quad {\rm in}\ \ Ω\setminus\{0\},\qquad u\geq 0\quad{\rm in}\ \ \mathbb{R}^N\setminusΩ,\qquad\qquad\quad $$ where $s\in(0,1)$, $Ω$ is a bounded domain containing the origin in $\mathbb{R}^N$ with $N>2s$ and $\frac{N}{N-2s}$ is the Serrin's critical exponent. We use an initial asymptotic at infinity to transform the critical case into a subcritical case where the underlying equation involves the fractional Hardy operator. The construction of singular solutions is based on the fact that some special functions are subsolutions of the original problem near the origin. We also classify the non-removable singularities of the solutions of $(E)$ and show the existence of a sequence of isolated singular solutions parameterizing the coefficients of the second order blow up term. To the best of our knowledge, this idea has also been first used in this work.

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arXiv2022

Discrete Schwarz rearrangement in lattice graphs

In this paper, we prove a discrete version of the generalized Riesz inequality on $\mathbb{Z}^d$. As a consequence, we will derive the extended Hardy-Littlewood and Pólya-Szegö inequalities. We will also establish cases of equality in the latter. Our approach is totally novel and self-contained. In particular, we invented a definition for the discrete rearrangement in higher dimensions. Moreover, we show that the definition "suggested" by Pruss does not work. We solve a long-standing open question raised by Alexander Pruss in [Pru98, p494], Duke Math Journal, and discussed with him in several communications in 2009-2010, [Pru10]. Our method also provides a line of attack to prove other discrete rearrangement inequalities and opens the door to the establishment of optimizers of many important discrete functional inequalities in $\mathbb{Z}^d,$ $d\geq2$. We will also discuss some applications of our findings. To the best of our knowledge, our results are the first ones in the literature dealing with discrete rearrangement on $\mathbb{Z}^d,$ $d\geq2$.

math.AP↗
arXiv2022

Threshold for Existence, Non-existence and Multiplicity of positive solutions with prescribed mass for an NLS with a pure power nonlinearity in the exterior of a ball

We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent p = 6 when N = 2, which does not seem to have played a role for this equation in the past. Moreover, our findings are "quite surprising" and completely different from the results obtained on the entire space and on balls. We will also show that the nature of the domain is crucial for the existence and stability of standing waves. As a foretaste, it is well-known that in the supercritical case these waves are unstable in RN . In this paper, we will show that in the exterior domain they are strongly stable.

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