Avancées en Mathématiques Pures et Appliquées

APAM - ISSN 1869-6090 - © ISTE Ltd

Numéros parus

Derniers articles parus

We consider a compact Riemannian manifold $$$(M, g)$$$ of dimension $$$N \geq 3$$$ and $$$\Sigma$$$ a closed totally geodesic submanifold of dimension $$$1 \leq k \leq N-2$$$, and $$$h: M \to ℝ$$$ is a continuous function such that the linear operator $$$-Δ_g+h$$$ is coercive. We study existence of positive solutions $$$u \in H^1\left(M\right)$$$ to the following nonlinear PDE with two Hardy-Sobolev critical exponents :
(0.1) $$$ -\Delta_g u+h u=\lambda \rho_{\Sigma}^{-s_1} u^{2^*_{s_1}-1}+\rho_{\Sigma}^{-s_2} u^{2^*_{s_2}-1} \qquad \textrm{ in } (M, g)$$$
where $$$\lambda$$$ is a positive parameter, $$$0 < s_2 < s_1 < 2$$$, the $$$2^*_{s_i}:=\frac{2(N-s_i)}{N-2}$$$ $$$(i=1, 2)$$$ are two critical Hardy-Sobolev exponents and $$$\rho_\Sigma: \mathcal{M} \to ℝ$$$ is the distance function to $$$\Sigma$$$. In this paper, we give sufficient condition depending on the local geometries of the submanifold $$$\Sigma$$$ and the manifold $$$M$$$, for the existence of mountain pass solution to (0.1).

The main objective of this paper is to characterize a class of lattices, called splittable lattices, in the nilpotent semidirect product Lie group $$$G=\mathbb R^n\rtimes_\eta\mathbb R^m$$$. Here, $$$\eta$$$ denotes the representation of the abelian Lie group $$$\mathbb R^m$$$ in $$$GL_n(\mathbb R)$$$ defined by $$$\eta(t_{1},\dots,t_{m})=\exp{(\sum_{i=1}^mt_iN_i)}$$$ with $$$(N_i)_{1\leq i\leq m}$$$ is a set of pairwise commuting nilpotent matrices in $$$\mathbb Q^{n\times n}$$$. As an application we study the case of the nilpotent Lie group $$$F_{n,m}=\mathbb R^n\rtimes_\eta\mathbb R^m$$$ where $$$N_i=J^i$$$ with $$$J=\sum_{k=1}^{n-1}E_{k+1,k}$$$ where $$$E_{k+1,k}\in\mathbb R^{n\times n}$$$ denotes the matrix given by the Kronecker delta, namely, $$$\big(E_{k+1,k}\big)_{i,j}=\delta_{i,k+1}\delta_{j,k}$$$.

The study of manifolds equipped with special linear connections has become an important topic in contemporary Riemannian geometry. In this work, we investigate Kenmotsu manifolds endowed with a class of semi-symmetric non-metric connections that generalize several previously known connections in the literature. Particular attention is devoted to the geometric behavior of symmetric and skew-symmetric parallel tensor fields associated with such connections. We further examine the conditions under which these manifolds satisfy Ricci-symmetric, weakly symmetric, and weakly Ricci-symmetric properties. In addition, the existence and structure of Ricci solitons related to these non-metric connections are studied within the setting of Kenmotsu geometry.

This work deals with traveling waves solutions of bistable reaction-diffusion systems in infinite cylinders. It is known that unique, stable traveling fronts exist for bistable systems. Here we are concerned with the traveling wave speed and how small perturbations affect the speed of the original problem. Using a mini-max variational formula, we prove an asymptotic expansion of the traveling wave speed and we determine the exact value of the first-order coefficient of this expansion.

This work aims to present a novel fashion of the multi-dimensional cubic functional equations and to investigate its Hyers-Ulam stability in the setting of Banach spaces on the restricted domains under a condition as (C). Moreover, a subset $$$\Omega\subset\mathbb R^d$$$ of Lebesgue measure zero satisfying the mentioned condition is constructed. Lastly, applying this construction and by means of the Baire category theorem, an asymptotic behavior for an approximate solution of the d-dimension cubic functional equations on real numbers, is presented.