Multiple solutions for perturbed quasilinear elliptic problems

We investigate the existence of multiple solutions for the $(p,q)$-quasilinear elliptic problem \[ \begin{cases} -\Delta_p u -\Delta_q u\ =\ g(x, u) + \varepsilon\ h(x,u) & \mbox{in } \Omega,\\ u=0 & \mbox{on } \partial\Omega,\\ \end{cases} \] where $1< p< q< +\infty$, $\Omega$ is an open bounded domain of ${\mathbb R}^N$, the nonlinearity $g(x,u)$ behaves at infinity as $|u|^{q-1}$, $\varepsilon\in{\mathbb R}$ and $h\in C(\overline\Omega\times{\mathbb R},{\mathbb R})$. In spite of the possible lack of a variational structure of this problem, from suitable assumptions on $g(x,u)$ and appropriate procedures and estimates, the existence of multiple solutions can be proved for small perturbations.