In this paper, we are interested in positive solutions of

$$ \begin{align*}\left\{ \begin{array}{@{}ll} -\Delta u = a(x)v^{p-1}, \quad &\text{ in } \Omega,\\ -\Delta v = b(x)u^{q-1}, \quad &\text{ in } \Omega,\\ u,v>0, \quad &\text{ in } \Omega,\\ u=v=0, \quad &\text{ on } \partial\Omega, \end{array} \right. \end{align*} $$

$\Omega $

${\mathbb {R}}^N (N \ge 3)$

$ a(x), b(x)$

$\Bigl \lfloor \frac {N}{2} \Bigr \rfloor $

$\frac {N}{2}$

$ a(x)=b(x)=1$

$\Omega $

$$ \begin{align*}(p-1)(q-1)> \Big(1+\frac{2N}{\lambda_H}\Big)^2\left(\frac{q}{p}\right),\end{align*} $$

$\lambda _H$

$\Omega .$

$\lambda _H$

$q.$

Variational principles are derived in order to facilitate the investigation of the vibrations and stability of single and double-walled carbon nanotubes conveying a fluid, from a linear time-dependent partial differential equation governing their displacements. The nonlocal elastic theory of Euler-Bernoulli beams takes small-scale effects into account. Hamilton’s principle is obtained for double-walled nano-tubes conveying a fluid. The natural and geometric boundary conditions identified are seen to be coupled and time-dependent due to nonlocal effects.

A variational principle is established to provide a new formulation for convex Hamiltonian systems. Using this formulation, we obtain some existence results for second-order Hamiltonian systems with a variety of boundary conditions, including nonlinear ones.

This paper examines the strengths and weaknesses of one of science's most pervasive and flexible metaprinciples; optimality is used to explain utility maximization in economics, least effort principles in physics, entropy in chemistry, and survival of the fittest in biology. Fermat's principle of least time involves both teleological and causal considerations, two distinct modes of explanation resting on poorly understood psychological primitives. The rationality heuristic in economics provides an example from social science of the potential biases arising from the extreme flexibility of optimality considerations, including selective search for confirming vidence, ex post rationalization, and the confusion of prediction with explanation. Commentators are asked to reflect on the extent to which optimality is (1) an organizing principle of nature, (2) a set of relatively unconnected techniques of science, (3) a normative principle for rational choice and social organization, (4) a metaphysical way of looking at the world, or (5) something else still.

Cet article présente une comparaison de quatreconfigurations d'essais hétérogènes en vue d'optimiserl'identification des paramètres de comportement d'un matériauorthotrope élastique. La technique d'identification est basée surla minimisation d'une norme énergétique formulée sur la base del'erreur en relation de comportement. La norme est construite surla base de principes variationnels. Elle exploite les champs dedéformations mesurés au moyen d'une technique de mesure de champscinématiques et les déformations calculées par le code d'élémentsfinis Abaqus. Le problème de minimisation est résolu itérativementpar un algorithme régularisé : Levenberg-Marquardt. Une analyse desensibilité aux bruits de mesure est réalisée afin d'illustrer larobustesse et la stabilité numérique. Deux types de bruit simulantles erreurs de mesures, ont été testés : aléatoire etsystématique. Il ressort de cette étude que la configurationd'essai notée géométrie complexe donne le meilleur résultatd'identification.