‎let $g$ be a graph without an isolated vertex‎, ‎the normalized laplacian matrix $tilde{mathcal{l}}(g)$‎‎is defined as $tilde{mathcal{l}}(g)=mathcal{d}^{-frac{1}{2}}mathcal{l}(g) mathcal{d}^{-frac{1}{2}}$‎, where ‎$‎mathcal{‎d}‎$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎g‎$‎‎. ‎the eigenvalues of‎‎$tilde{mathcal{l}}(g)$ are ‎called ‎ ‎ as ‎the ‎normalized laplacian ...

<abstract><p>In an earlier paper, Cruz-Uribe, Rodney and Rosta proved equivalence between weighted Poincaré inequalities the existence of weak solutions to a family Neumann problems related degenerate $ p $-Laplacian. Here we prove similar in variable exponent spaces {p(\cdot)} $-Laplacian, non-linear elliptic equation with nonstandard growth conditions.</p></abstract>

In this work we apply the unfolding operator method to analyze asymptotic behavior of solutions $p$-Laplacian equation with Neumann boundary condition set in a bounded thin domain type $R^\varepsilon=\left\lbrace(x,y)\in\mathbb{R}^2:x\in(0,1)\mbox{ and }0<y<\varepsilon g\left({x}/{\varepsilon^\alpha}\right)\right\rbrace$ where $g$ is positive periodic function. We study three cases $0<\alpha<1$...