We study the boundary value problem $-{\mathrm div}((|\nabla u|^{p_1(x) -2}+|\nabla u|^{p_2(x)-2})\nabla u)=f(x,u)$ in $\Omega $, $u=0$ on $\partial \Omega $, where $\Omega $ is a smooth bounded domain in ${\mathbb{R}} ^N$. Our attention is focused on two cases when $f(x,u)=\pm (-\lambda |u|^{m(x)-2}u+|u|^{q(x)-2}u)$, where $m(x)=\max \lbrace p_1(x),p_2(x)\rbrace $ for any $x\in \overline{\Omega }$ or $m(x)<q(x)< \frac{N\cdot m(x)}{(N-m(x))}$ 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 functionals of the Mountain Pass Theorem and some adequate variational methods.
In this paper I seek to contribute to our understanding of Romanian immigration in the Czech Republic that has been neglected so far in the scholarly literature. This article presents evidence on a selected Romanian community in peripheral village in the Jeseník-region based upon biographic interviews. The emergence and function of the community are reflected in the migration history of Romanian families, their gradual integration in society and in place of residence (marginalized village). Special emphasis is placed upon crucial moments like emergence of the distinction between Romanians and Roma in the eyes of majority of the society. Positive as well as negative influences upon the integration are discussed. The paper shows empirical evidence which is in some aspects contradictory to findings about integration of foreigners in the Czechia as presented by Tollarová (2006). The paper concludes with some consideration about future development of the community.