An existence result for perturbed (p,q)-quasilinear elliptic problems

We investigate the existence of solutions of the ${(p,q)-}$quasilinear elliptic problem \[\left\{\begin{array}{lll} \displaystyle{ -\Delta_p u -\Delta_q u\ =\ g(x, u) + \varepsilon\, h(x,u)} & & \mbox{ in } \Omega,\\ \displaystyle{u=0} & & \mbox{ on } \partial\Omega,\\ \end{array}\right.\] where $\Omega$ is an open bounded domain in $\R^N$, $1<+\infty$, the nonlinearity $g(x,u)$ behaves at infinity as $|u|^{q-1}$, $\varepsilon\in\R$ and $h\in C(\overline\Omega\times\R,\R)$. In spite of the possible lack of a variational structure of this problem, appropriate procedures and estimates allow us to prove the existence of at least one nontrivial solution for small perturbations.