We study the boundary value problem $-{\mathrm{d}iv\left(\right(\left|\nabla u\right|}^{p_1\left(x\right)-2}+{\left|\nabla u\right|}^{p_2\left(x\right)-2}\left)\nabla u\right)=f(x,u)$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in ${ℝ}^N$. Our attention is focused on two cases when $f(x,u)=\pm (-\lambda |u{|}^{m\left(x\right)-2}u+\left|u{|}^{q\left(x\right)-2}u\right)$, where $m\left(x\right)=max\{p_1\left(x\right),p_2\left(x\right)\}$ for any $x\in \overline{\Omega }$ or $m\left(x\right)<q\left(x\right)<\frac{N·m\left(x\right)}{(N-m(x\left)\right)}$ for any $x\in \overline{\Omega }$. In the former case we show the existence of infinitely many weak solutions for any $\lambda >0$. In the latter we prove that if $\lambda$ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ${\mathbb{Z}}_2$-symmetric version for even...

We study the nonlinear boundary value problem involving reflection of the argument $$-M\left({\int }_{-1}^1{\left|u^{\text{'}}\left(s\right)\right|}^{2}ds\right)u^{\text{'}\text{'}}\left(x\right)=f(x,u\left(x\right),u(-x))x\in [-1,1],$$ where $M$ and $f$ are continuous functions with $M>0$. Using Galerkin approximations combined with the Brouwer’s fixed point theorem we obtain existence and uniqueness results. A numerical algorithm is also presented.

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $\left({\nabla }_{\alpha }{u}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_3{u}_{\epsilon }\right)$ bounded in $L^p(\Omega ;{ℝ}^9),1\&lt;p\&lt;+\infty .$ Here it is shown that, up to a subsequence, $u_{\epsilon }$ may be decomposed as $w_{\epsilon }+z_{\epsilon },$ where $z_{\epsilon }$ carries all the concentration effects, i.e. $\left\{{\left|\left({\nabla }_{\alpha }{w}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_3{w}_{\epsilon }\right)\right|}^p\right\}$ is equi-integrable, and $w_{\epsilon }$ captures the oscillatory behavior, i.e. $z_{\epsilon }\to 0$ in measure. In addition, if $\left\{u_{\epsilon }\right\}$ is a recovering sequence then $z_{\epsilon }=z_{\epsilon }\left(x_{\alpha }\right)$ nearby $\partial \Omega .$