arXiv subjects
Publications and source records attributed to Hichem Hajaiej.
We are concerned with existence results for a critical problem of Brezis-Nirenberg Type involving an integro-differential operator. Our study includes the fractional Laplacian. Our approach still applies when adding small singular terms. It hinges on appropriate choices of parameters in the mountain-pass theorem
The main goal of this paper is the study of two kinds of nonlinear problems depending on parameters in unbounded domains. Using a nonstandard variational approach, we first prove the existence of bounded solutions for nonlinear eigenvalue problems involving the fractional Laplace operator and nonlinearities that have subcritical growth. In the second part, based on a variational principle of Ricceri [16], we study a fractional nonlinear problem with two parameters and prove the existence of multiple solutions.
The purpose of this article is to give a complete study of the weak solutions of the fractional elliptic equation \begin{equation}\label{00} \arraycolsep=1pt \begin{array}{lll} (-Δ)^α u+u^p=0\ \ \ \ &\ {\rm in}\ \ B_1(e_N),\\[2mm]\phantom{(-Δ)^α +u^p} u=δ_{0}& \ {\rm in}\ \ \mathbb{R}^N\setminus B_1(e_N), \end{array} \end{equation} where $p>0$, $ (-Δ)^α$ with $α\in(0,1)$ denotes the fractional Laplacian operator in the principle value sense, $B_1(e_N)$ is the unit ball centered at $e_N=(0,\cdots,0,1)$ in $\mathbb{R}^N$ with $N\ge 2$ and $δ_0$ is the Dirac mass concentrated at the origin. We prove that problem (\ref{00}) admits a unique weak solution when $p> 1+\frac{2α}{N}$. Moreover, if in addition $p\ge \frac{N+2}{N-2}$, the weak solution vanishes as $α\to 1^-$. We also show that problem (\ref{00}) doesn't have any weak solution when $p\in[0, 1+\frac{2α}{N}]$. These results are very surprising since there are in total contradiction with the classical setting, i.e. $$ \arraycolsep=1pt \begin{array}{lll} -Δu+ u^p=0\ \ \ \ &\ {\rm in}\ \B_1(e_N),\\[2mm] \phantom{-Δ+u^{p} } u=δ_{0}& \ {\rm in}\ \ \R^N\setminus B_1(e_N), \end{array} $$ for which it has been proved that there are no solutions for $p\ge \frac{N+1}{N-1}$.
Let $α\in(0,1)$, $Ω$ be a bounded open domain in $R^N$ ($N\ge 2$) with $C^2$ boundary $\partialΩ$ and $ω$ be the Hausdorff measure on $\partialΩ$. We denote by $\frac{\partial^αω}{\partial \vec{n}^α}$ a measure $$\langle\frac{\partial^αω}{\partial \vec{n}^α},f\rangle=\int_{\partialΩ}\frac{\partial^αf(x)}{\partial \vec{n}_x^α} dω(x),\quad f\in C^1(\barΩ),$$ where $\vec{n}_x$ is the unit outward normal vector at point $x\in\partialΩ$. In this paper, we prove that problem $$ \begin{array}{lll} (-Δ)^αu+g(u)=k\frac{\partial^αω}{\partial \vec{n}^α}\quad & {\rm in}\quad \barΩ,\\[2mm] \phantom{(-Δ)^α+g(u)} u=0\quad & {\rm in}\quad Ω^c \end{array} $$ admits a unique weak solution $u_k$ under the hypotheses that $k>0$, $(-Δ)^α$ denotes the fractional Laplacian with $α\in(0,1)$ and $g$ is a nondecreasing function satisfying extra conditions. We prove that the weak solution is a classical solution of $$ \begin{array}{lll} \ \ \ (-Δ)^αu+g(u)=0\quad & {\rm in}\quad Ω,\\[2mm] \phantom{------\} \ u=0\quad & {\rm in}\quad R^N\setminus\barΩ,\\[2mm] \phantom{} \lim_{x\inΩ,x\to\partialΩ}u(x)=+\infty. \end{array} $$
The purpose of this paper is to study the weak solutions of the fractional elliptic problem \begin{equation}\label{000} \begin{array}{lll} (-Δ)^αu+εg(u)=k\frac{\partial^αν}{\partial \vec{n}^α}\quad &{\rm in}\quad\ \ \barΩ,\\[3mm] \phantom{(-Δ)^α+εg(u)} u=0\quad &{\rm in}\quad\ \ \barΩ^c, \end{array} \end{equation} where $k>0$, $ε=1$ or $-1$, $(-Δ)^α$ with $α\in(0,1)$ is the fractional Laplacian defined in the principle value sense, $Ω$ is a bounded $C^2$ open set in $R^N$ with $N\ge 2$, $ν$ is a bounded Radon measure supported in $\partialΩ$ and $\frac{\partial^αν}{\partial \vec{n}^α}$ is defined in the distribution sense, i.e. $$ \langle\frac{\partial^αν}{\partial \vec{n}^α},ζ\rangle=\int_{\partialΩ}\frac{\partial^αζ(x)}{\partial \vec{n}_x^α}dν(x), \qquad \forallζ\in C^α(R^N), $$ here $\vec{n}_x$ denotes the unit inward normal vector at $x\in\partialΩ$.
The purpose of this paper is to extend the embedding theorem of Sobolev spaces involving general kernels and we provide a sharp critical exponent in these embeddings. As an application, solutions for equations driven by a general integro-differential operator, with homogeneous Dirichlet boundary conditions, is established by using the Mountain Pass Theorem.
We study the stability of the standing wave solutions of a Gross-Pitaevskii equation describing Bose-Einstein condensation of dipolar quantum gases and characterize their orbit. As an intermediate step, we consider the corresponding constrained minimization problem and establish existence, symmetry and uniqueness of the ground state solutions.
In this paper, we prove the existence of minimizers of a class of multi-constrained variational problems. We consider systems involving a nonlinearity that does not satisfy compactness, monotonicity, neither symmetry properties. Our approach hinges on the concentration-compactness approach. In the second part, we will treat orthogonal constrained problems for another class of integrands using density matrices method.
We Proveexistence, symmetry and uniqueness of standing waves of a BCE with josephson junction. We also characterize the orbit of standing waves
We show the existence of ground state and orbital stability of standing waves of fractional Schrödinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.
We study the Cauchy problem for the fractional Schrödinger equation $$ i\partial_tu = (m^2-Δ)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < α< 2$, and $F$ stands for the nonlinearity of Hartree type: $$F(u) = λ(\frac{ψ(\cdot)}{|\cdot|^γ} * |u|^2)u$$ with $λ= \pm1, 0 <γ< n$, and $0 \le ψ\in L^\infty(\mathbb R^n)$. We prove the existence and uniqueness of local and global solutions for certain $α$, $γ$, $λ$, $ψ$. We also remark on finite time blowup of solutions when $λ= -1$.
We analyse a nonlinear Schrödinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree-Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an X-ray Free Electron Laser (XFEL). We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding solutions of a Schrödinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential.
We establish general assumptions under which a constrained vari- ational problem involving the fractional gradient and a local nonlin- earity admits minimizers.
In this paper we prove the Fractional Gagliardo-Nirenberg Inequality, Polya-Szego Inequality and the Sharp Fractional Sobolev Inequality, we then provide an application of such inequalities in a constraiend variational problem involving the fractional gradient and alocal non-linearity
Sufficient and necessary conditions for the generalized Gagliardo-Nirenberg (GN) inequality in Besov spaces and Triebel-Lizorkin spaces are obtained. Applying the GN inequality, we show that the finite-time blowup solutions have concentration phenomena in critical Lebesgue space L^3. Moreover, we consider the minimizer for a class of variational problem by applying the fractional GN inequality.
We prove a weak-strong convergence result for functionals of the form $\int_{\mathbb{R}^N} j(x, u, Du)\,dx$ on $W^{1,p}$, along equiintegrable sequences. We will then use it to study cases of equality in the extended Polya-Szegö inequality and discuss applications of such a result to prove the symmetry of minimizers of a class of variational problems including nonlocal terms under multiple constraints.
We study well-posedness, local and global, existence of solutions for a general class of physically meaningful nonlinear Schrödinger systems with magnetic field involving local and nonlocal nonlinearities.
In this second part, we establish the existence of special solutions of the nonlinear Schrödinger system studied in the first part when the diamagnetic field is nul. We also prove some symmetry properties of these ground states solutions.