arXiv subjects

Hichem Hajaiej

Publications and source records attributed to Hichem Hajaiej.

At least 19 recordsLinked to original sources

arXiv2026

Transverse dependence and asymptotic dimensional reduction of ground states for the logarithmic Schrödinger equation on a waveguide manifold

In this paper, we develop a novel approach to studying the $y$-dependence of ground states for the logarithmic Schrödinger equation on $\mathbb{R}^d \times \mathbb{T}$. Unlike in \cite{STNT2014} (Analysis \& PDE, 2014) and \cite{HH1} (SIAM J. Math. Anal., 2024), the power-law scaling argument does not transfer directly to the logarithmic setting. Instead, amplitude scaling gives an exact identity showing that transverse dependence is independent of the prescribed mass. We also discuss uniqueness for the Cauchy problem and conditional orbital stability.

math.AP↗

arXiv2026

A new class of nonlocal operators: the fractional logarithmic $p$-Laplacian and its properties

In this paper, we introduce and investigate the fractional logarithmic $p$-Laplacian $(-Δ)_{p}^{s+\log}$, defined as the first-order derivative with respect to the parameter $t$ of the fractional $p$-Laplacian $(-Δ)_{p}^{t}$ evaluated at $t=s$. We establish that this operator admits the following integral representation \[ \begin{aligned} (-Δ)_{p}^{s+\log} u(x) &= B(N,s,p)(-Δ)_{p}^{s}u(x)\\ &\quad -p\, C(N,s,p)\mathrm{P.V.}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ln |x-y|}{|x-y|^{N+sp}}dy, \end{aligned} \] where $C(N,s,p)$ denotes the standard normalization constant associated with the fractional $p$-Laplacian, and $B(N,s,p)=\frac{d}{ds}\left(\ln C(N,s,p)\right)$. As a consequence of this representation, it follows that the operator is nonlocal and of logarithmic type, and may be viewed as a nonlinear analogue of the fractional logarithmic Laplace operator recently introduced by Chen et al. \cite{Chen-Chen-Hauer}. We further develop the associated functional framework in both $\mathbb{R}^{N}$ and bounded Lipschitz domains by introducing the natural energy spaces adapted to problems driven by $(-Δ)_{p}^{s+\log}$. Within this framework, fundamental functional inequalities are established, in particular Pohozaev-type identities and D\'ıaz-Saa inequalities, which are of independent interest and applicable to a broader class of problems. Moreover, we derive results concerning density, continuity, and compact embedding properties. We emphasize that the compactness of the embedding is proved at the critical exponent $p^{*}_{s}=\frac{Np}{N-sp}$, which distinguishes the present setting from the classical Sobolev and fractional Sobolev frameworks. Finally, as an application, we investigate the associated Dirichlet eigenvalue problem and derive existence, uniqueness, and boundedness results for the corresponding solutions.

math.AP↗

arXiv2026

Lower and Upper Bounds for Sums of Eigenvalues of the Fractional-Logarithmic Laplacian

In this work, we establish lower and upper bounds for sums of the Dirichlet eigenvalues of the fractional-logarithmic Laplacian. A main challenge in such a study comes from the fact that this operator has a Fourier symbol that is not globally monotone in its radial variable due to its low-frequency behavior. The bounds have the correct leading-order asymptotic. To complement the theoretical analysis, we also present a one-dimensional finite element approximation of the fractional-logarithmic eigenvalue problem. We compare the computed eigenvalue sums with the explicit lower bound and the principal asymptotic term

math.AP↗

arXiv2026

Existence of positive and sign-changing solutions for a Choquard equation involving mixed local and nonlocal operators

We study the Choquard equation involving mixed local and nonlocal operators \[ -Δu + (-Δ)^{s}u + V(x)u = \left(\frac{1}{|x|^μ} * F(u)\right) f(u) \quad \text{in } \mathbb{R}^{2}, \] where $s\in(0,1)$, $μ\in(0,2)$, $F(t)=\int_{0}^{t} f(τ)\,dτ$, and $f$ has subcritical exponential growth of Trudinger--Moser type. Under suitable assumptions on the potential $V$ and the nonlinearity $f$, we prove the existence of a least energy positive solution by a Nehari manifold approach. We also establish the existence of a sign-changing solution by means of invariant sets of descending flow. If, in addition, the nonlinearity is odd, then the problem admits infinitely many sign-changing solutions.

math.AP↗

arXiv2026

The best constant in the G-N inequality for the mixed local and Nonlocal Laplacian

In this paper, we establish the best constant in the G-N inequality for the mixed local and nonlocal Laplacian. In our problem, classical methods cannot apply directly since regularity results for the operator under study seem to be highly challenging. We build an innovative method that not only enabled us to prove that the optimizer of the best constant is the ground state solution of the equation, but also to establish the best constant.

math.AP↗

arXiv2026

Local Existence, Uniqueness, Regularity, and Global Behavior of Evolution Equations Involving Mixed Local and Nonlocal Operators

In this work, we address a parabolic problem featuring a potentially doubly nonlinear term, governed by a combination of local and nonlocal operators (see Problem P1 below). We first establish the local existence of weak energy solutions via a semidiscretization in time applied to an auxiliary evolution problem. The uniqueness of these solutions is subsequently obtained through a novel generalization of the classical inequality of Diaz and Saa, suitably adapted to the mixed local nonlocal setting. This generalization provides a new comparison principle and establishes the T-accretivity of a corresponding operator in L2. By employing this comparison principle, we construct suitable barrier functions that allow the global in time extension of solutions. Furthermore, we demonstrate the convergence of weak solutions to a nontrivial stationary state. Our approach relies on methods from the theory of contraction semigroups. It is noteworthy that these results are underpinned by a detailed analysis of the stationary problems associated with Problem P1, which also reveals several qualitative properties of the solutions.

math.AP↗

arXiv2026

The role of the mean curvature in nonlinear p-Laplacian problems with critical exponent

We deal with critical nonlinear problems involving the p-Laplacian operator on bounded domains with mixed boundary conditions. We prove the existence of least energy solutions. Our work shows a significant difference between the semi-linear case p = 2 and the quasilinear case for the existence results. Moreover, neither the results for the Laplacian can be extended to the p-Laplacian, nor the method for the p-Laplacian can apply to the Laplacian setting. Additionally, the cases (p < 2 and p > 2) present different challenges and need to be studied separately. More precisely, when p > 2, the effect of the geometry of the boundary conditions dominates that one of the potential, whereas for p < 2 the opposite behavior holds true.

math.DG↗

arXiv2026

Ground State Solutions For Local-Nonlocal Shrodinger Equations In the Presence Of Two Critical Exponents

In this paper, we address the existence of ground state solutions for Schrodinger equations in the presence of local and nonlocal operators and two critical nonlinearities associated with each operator. The situation is completely solved in the critical case both in the local and in the nonlocal settings. However, methods developed in these cases cannot extend to local-nonlocal operators when we have two critical power nonlinearities. This unprecedented situation in PDEs presents some challenges, and its resolution will open the door to solve similar problems. Our approach is essentially based on a subtle generalization of the Lieb translation theorem. We will also discuss the positivity of the ground states as well as their regularity.

math.AP↗

arXiv2026

Explicit Formula Of The Critical Mass And The Energy Ground State Solution For The Mixed Local-Nonlocal Schrodinger Equation For The In-Between Critical Exponents Case

Our first main contribution consists in establishing an explicit formula of the critical mass via the best constant of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal Laplacian. We also prove the existence of an optimizer of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal operator. We then show that the optimizer (after some suitable scaling) is an energy ground state solution (with critical mass ). This is a key ingredient to determine sufficient and necessary conditions of existence and non-existence of energy ground state solutions in the in-between critical exponents case. Finally, we show that the energy ground state solution uc0 is an optimizer of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal operator.

math.AP↗

arXiv2025

Nonlocal Dirichlet problems involving the Logarithmic $p$-Laplacian

In this work, we show the existence of an unbounded sequence of minimax eigenvalues for the logarithmic $p$-Laplacian via the $\mathbb{Z}_2$-cohomological index of Fadell and Rabinowitz. As an application of these minimax eigenvalues and $p$-logarithmic Sobolev inequality proved in [4], we prove new existence results for nonlocal Dirichlet problems involving logarithmic $p$-Laplacian and nonlinearities with $p$-superlinear and subcritical growth at infinity.

math.AP↗

arXiv2025

Sharp embeddings and existence results for Logarithmic $p$-Laplacian equations with critical growth

In this paper, we derive a new $p$-Logarithmic Sobolev inequality and optimal continuous and compact embeddings into Orlicz-type spaces of the function space associated with the logarithmic $p$-Laplacian. As an application of these results, we study a class of Dirichlet boundary value problems involving the logarithmic $p$-Laplacian and critical growth nonlinearities perturbed with superlinear-subcritical growth terms. By employing the method of the Nehari manifold, we prove the existence of a nontrivial weak solution. Lastly, we conduct an asymptotic analysis of a weighted nonlocal, nonlinear problem governed by the fractional $p$-Laplacian with superlinear or sublinear type non-linearity, demonstrating the convergence of least energy solutions to a non-trivial, non-negative least energy solution of a Brezis-Nirenberg type or logistic-type problem, respectively, involving the logarithmic $p$-Laplacian as the fractional parameter $s \to 0^+$. The findings in this work serve as a nonlinear analogue of the results reported in \cite{Angeles-Saldana, Arora-Giacomoni-Vaishnavi, Santamaria-Saldana}, thereby extending their scope to a broader variational framework.

math.AP↗

arXiv2025

Non variational type critical growth nonlocal system

This study investigates the existence, uniqueness, and multiplicity of positive solutions for a system of fractional differential equations given by: \begin{equation*} (-Δ)^{s_i} u_{i}+λ_{i} u_{i}=\sum_{j=1}^{n} α_{i j}\left|u_{j}\right|^{q_{i j}}\left|u_{i}\right|^{p_{i j}-2} u_{i} , u_i\in {\mathscr{D}^{s_i,2}\left(\mathbb{R}^{N}\right)}, i=1,2,\cdots,n, \end{equation*} where $N>2s=\max\{2s_i\}$, $s_i\in(0,1)$, $n\geq 2$, $λ_{i} \geq 0$, $α_{ij}>0$, $p_{ij}<2^{*}_{s}$, and $p_{ij}+q_{ij}=2^{*}_{s}=\min\{{\frac{2N}{N-2s_i}\}}$ for $i\neq j \in \{1,2,...,n\}$. $2^{*}_s$ called the fractional critical sobolev exponent and $2^{*}_s=2 N /(N-2s)$ for $N > 2s$ and $2^{*}_s=+\infty$ for $N=2s$ or $N<2s$. Our work establishes novel uniqueness and multiplicity results for positive solutions, applicable whether the system possesses a variational structure or not. We provide a comprehensive characterization of the exact number of positive solutions under specific parameter configurations. Our analysis shows that the positive solution set behaves differently across three distinct regimes: $p_{ij}<2$, $p_{ij}=2$, and $2<p_{ij}<2^{*}_{s}$.

math.AP↗

arXiv2025

The mixed fractional Hartree equations in Fourier amalgam and modulation spaces

We prove local and global well-posedness for mixed fractional Hartree equation and with low regularity Cauchy data in Fourier amalgam $\F W(L^p,\ell^q)$ and modulation $M^{p,q}$ spaces. Similar results also hold for the Hartree equation with harmonic potential in some modulation spaces. Our approach also addresses Hartree-Fock equations of finitely many (but arbitrary large) particles. A key ingredient of our method is to establish trilinear estimates for Hartree non-linearity and the use of Strichartz estimates. As a consequence, we could gain $\F W(L^p,\ell^q)$ and $M^{p,q}-$regularity for all $p,q\in [1, \infty].$ In particular, we extend result of Bhimani-Grillakis-Okoudju \cite{bhimani2020hartree} in $M^{p,q}$ for all $p,q$ and complement known results in Sobolev spaces.

math.AP↗

arXiv2025

Uniqueness of normalized ground states for NLS models

We present two methods to prove the uniqueness of normalized ground states. We will first discuss the key ideas and ingredients of each method. Then, we will apply them to various classes of PDEs. Our approach is applicable to other operators, domains and nonlinearities provided that some hypotheses are satisfied.

math.AP↗

arXiv2024

Normalized solutions to nonlinear Schrödinger equations with competing Hartree-type nonlinearities

In this paper, we consider solutions to the following nonlinear Schrödinger equation with competing Hartree-type nonlinearities, $$ -Δu + λu=\left(|x|^{-γ_1} \ast |u|^2\right) u - \left(|x|^{-γ_2} \ast |u|^2\right) u\quad \mbox{in} \,\, \R^N, $$ under the $L^2$-norm constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $0<γ_2 < γ_1 <\min\{N, 4\}$ and $λ\in \R$ appearing as Lagrange multiplier is unknown. First we establish the existence of ground states in the mass subcritical, critical and supercritical cases. Then we consider the well-posedness and dynamical behaviors of solutions to the Cauchy problem for the associated time-dependent equations.

math.AP↗

arXiv2024

Existence of normalized ground state solution to a mixed Schrödinger system in a plane

In this paper, we establish the existence of positive ground state solutions for a class of mixed Schrödinger systems with concave-convex nonlinearities in $\mathbb{R}^2$, subject to $L^2$-norm constraints; that is, \[ \left\{ \begin{aligned} -\partial_{xx} u + (-Δ)_y^s u + λ_1 u &= μ_1 u^{p-1} + βr_1 u^{r_1-1} v^{r_2}, && -\partial_{xx} v + (-Δ)_y^s v + λ_2 v &= μ_2 v^{q-1} + βr_2 u^{r_1} v^{r_2-1}, && \end{aligned} \right. \] subject to the $L^2$-norm constraints: \[ \int_{\mathbb{R}^2} u^2 \,\mathrm{d}x\mathrm{d}y = a \quad \text{and} \quad \int_{\mathbb{R}^2} v^2 \,\mathrm{d}x\mathrm{d}y = b, \] where $(x,y)\in \mathbb{R}^2$, $u, v \geq 0$, $s \in \left(1/2, 1 \right)$, $μ_1, μ_2, β> 0$, $r_1, r_2 > 1$, the prescribed masses $a, b > 0$, and the parameters $λ_1, λ_2$ appear as Lagrange multipliers. Moreover, the exponents $p, q, r_1 + r_2$ satisfy: \[ \frac{2(1+3s)}{1+s} < p, q, r_1 + r_2 < 2_s, \] where $2_s = \frac{2(1+s)}{1-s}$. To obtain our main existence results, we employ variational techniques such as the Mountain Pass Theorem, the Pohozaev manifold, Steiner rearrangement, and others, consolidating the works of Louis Jeanjean et al. \cite{jeanjean2024normalized}.

math.AP↗

arXiv2024

Standing waves with prescribed mass for Schrodinger equations with competing Van Der Waals type potentials

We investigate standing waves with prescribed mass for a class of Schrodinger equations with competing Van Der Waals type potentials, arising in a model of non-relativistic bosonic atoms and molecules. By developing an approach based on a direct minimization of the energy functional on a new constrained manifold, we establish the existence of two normalized solutions for the corresponding stationary problem. One is a local minimizer with positive level and the other one is a global minimizer with negative level. Moreover, we find that the global minimizer is farther away from the origin than the local minimizer. Finally, we explore the relations between the ground state solution and the least action solution, and some dynamical behavior and scattering results are presented as well.

math.AP↗

arXiv2024

New insights into the solutions of a class of anisotropic nonlinear Schrödinger equations on the plane

In this paper, we study the following anisotropic nonlinear Schrödinger equation on the plane, \[ \begin{cases} {\rm i}\partial_t Φ+\partial_{xx} Φ-D_y^{2s} Φ+|Φ|^{p-2}Φ=0,&\quad (t,x,y)\in\mathbb{R} \times \mathbb{R}^2, Φ(x,y,0)=Φ_0(x,y),&\quad (x,y)\in\mathbb{R}^2, \end{cases} \] where $D_y^{2s}=\left(-\partial_{yy}\right)^s$ denotes the fractional Laplacian with $0<s<1$ and $2<p<\frac{2(1+s)}{1-s}$. We first study the existence of normalized solutions to this equation in the subcritical, critical, and supercritical cases. To this aim, regularity results and a Pohozaev type identity are necessary. Then, we determine the conditions under which the solutions blow up. Furthermore, we demonstrate the existence of boosted traveling waves when $s\geq1/2$ and their decay at infinity. Additionally, for the delicate case $s=1/2$, we provide a non-existence result of boosted traveling waves and we establish that there is no scattering for small data. Finally, we also study normalized boosted travelling waves in the mass subcritical case. Due to the nature of the equation, we do not impose any radial symmetry on the initial data or on the solutions.

math.AP↗