## Abstract.

We study the nonlinear boundary value problem $-{\mathrm{div}\left(\right(\left|u\left(x\right)\right|}^{{p}_{1}\left(x\right)-2}+{\left|u\left(x\right)\right|}^{{p}_{2}\left(x\right)-2}\left)u\left(x\right)\right)$ $\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}={V}_{1}{\left(x\right)\left|u\right|}^{q\left(x\right)-2}u-{V}_{2}\left(x\right){\left|u\right|}^{\left(x\right)-2}u$ in $$, $u=0$ on $$, where $$ is a bounded domain in ${}^{N}$ with smooth boundary, $$, $$ are positive real numbers, ${p}_{1}$, ${p}_{2}$, $q$, $$ are continuous functions on $\stackrel{}{}$, ${V}_{1}$ and ${V}_{2}$ are weight functions in generalized Lebesgue spaces ${L}^{{s}_{1}\left(x\right)}\left(\right)$ and ${L}^{{s}_{2}\left(x\right)}\left(\right)$, respectively, such that ${V}_{1}>0$ in an open set ${}_{0}$ with $|{}_{0}|>0$ and ${V}_{2}0$ on $$. We prove, under appropriate conditions that for any $>0$ there exists ${}_{*}$ sufficiently small such that for any $(0,{}_{*})$ the above nonhomogeneous quasilinear problem has a nontrivial positive weak solution. The proof relies on some variational arguments based on Ekeland's variational principle.

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