arXiv subjects
Publications and source records attributed to Lamine Mbarki.
We study an elliptic nonlocal problem driven by a source term which is a Radon measure. We prove the existence of a weak solution by a weak convergence method. In the process, we define a new function space which we name the Walker space. The question about the existence of infinitely many nontrivial solutions leads to an interesting conjecture.
We study a conformally invariant equation involving the Dirac operator and a non-linearity of convolution type. This non-linearity is inspired from the conformal Einstein-Dirac problem in dimension 4. We first investigate the compactness, bubbling and energy quantization of the associated energy functional then we characterize the ground state solutions of the problem on the standard sphere. As a consequence, we prove an Aubin-type inequality that assures the existence of solutions to our problem and in particular the conformal Einstein-Dirac problem in dimension 4. Moreover, we investigate the effect of a linear perturbation to our problem, leading us to a Brezis-Nirenberg type result.
In the present manuscript, we focus on a novel tri-nonlocal Kirchhoff problem, which involves the $p(x)$-fractional Laplacian equations of variable order. The problem is stated as follows: \begin{eqnarray*} \left\{ \begin{array}{ll} M\Big(σ_{p(x,y)}(u)\Big)(-Δ)^{s(\cdot)}_{p(\cdot)}u(x) =λ|u|^{q(x)-2}u\left(\int_Ø\frac{1}{q(x)} |u|^{q(x)}dx \right)^{k_1}+β|u|^{r(x)-2}u\left(\int_Ø\frac{1}{r(x)} |u|^{r(x)}dx \right)^{k_2} \quad \mbox{in }Ω, \\ u=0 \quad \mbox{on }\partialΩ, \end{array} \right. \end{eqnarray*} where the nonlocal term is defined as $$ σ_{p(x,y)}(u)=\int_{Ω\times Ω}\frac{1}{p(x,y)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+s(x,y)p(x,y)}} \,dx\,dy. $$ Here, $Ω\subset\mathbb{R}^{N}$ represents a bounded smooth domain with at least $N\geq2$. The function $M(s)$ is given by $M(s) = a - bs^γ$, where $a\geq 0$, $b>0$, and $γ>0$. The parameters $k_1$, $k_2$, $λ$ and $β$ are real parameters, while the variables $p(x)$, $s(\cdot)$, $q(x)$, and $r(x)$ are continuous and can change with respect to $x$. To tackle this problem, we employ some new methods and variational approaches along with two specific methods, namely the Fountain theorem and the symmetric Mountain Pass theorem. By utilizing these techniques, we establish the existence and multiplicity of solutions for this problem separately in two distinct cases: when $a>0$ and when $a=0$. To the best of our knowledge, these results are the first contributions to research on the variable-order $p(x)$-fractional Laplacian operator.
This paper deals with a class of nonlocal variable $s(.)$-order fractional $p(.)$-Kirchhoff type equations: \begin{eqnarray*} \left\{ \begin{array}{ll} \mathcal{K}\left(\int_{\mathbb{R}^{2N}}\frac{1}{p(x,y)}\frac{|φ(x)-φ(y)|^{p(x,y)}}{|x-y|^{N+s(x,y){p(x,y)}}} \,dx\,dy\right)(-Δ)^{s(\cdot)}_{p(\cdot)}φ(x) =f(x,φ) \quad \mbox{in }Ω, \\ φ=0 \quad \mbox{on }\mathbb{R}^N\backslashΩ. \end{array} \right. \end{eqnarray*} Under some suitable conditions on the functions $p,s, \mathcal{K}$ and $f$, the existence and multiplicity of nontrivial solutions for the above problem are obtained. Our results cover the degenerate case in the $p(\cdot)$ fractional setting.
In the context of continuous zooming systems $f:M \to M$ on a compact metric space $M$, which include the non-uniformly expanding ones, possibly with the presence of a critical set, with the zooming set dense in $M$, we prove that any H\"older potential $\phi : M \to \mathbb{R}$ for which the integrals $\int \phi d\mu \geq 0$ with respect to any $f$-invariant probability $\mu$, admits a continuous function $\lambda_{0} : M \to \mathbb{R}$ (which can be H\"older if some integral is positive) such that \[ \phi \geq \lambda_{0}- \lambda_{0} \circ f. \] This extends a result in [9] for $C^{1}$-expanding maps on the circle $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ to important classes of maps as uniformly expanding, local diffeomorphisms with non-uniform expansion, Viana maps, Benedicks-Carleson maps and Rovella maps. We also give an example beyond the exponential contractions context. Moreover, in the case of the integrals $\int \phi d\mu = 0$ with respect to any $f$-invariant probability $\mu$ and the set of periodic points to be dense in $M$, we obtain a version of the Livsic Theorem, that is, the functions $\lambda_{0}$ can be taken such that \[ \phi = \lambda_{0}- \lambda_{0} \circ f. \] Additionally, we also prove that the measure which maximizes the integrals is unique for a residual set of potentials.