Multiple solutions for perturbed quasilinear elliptic problems,

We investigate the existence of multiple solutions for the $(p,q)$--quasilinear elliptic problem \[\left\{\begin{array}{ll} -\Delta_p u -\Delta_q u\ =\ g(x, u) + \varepsilon\ h(x,u) & \mbox{ in } \Omega,\\ u=0 & \mbox{ on } \partial\Omega,\\ \end{array}\right.\] where $1<+\infty$, $\Omega$ is an open bounded domain of $\R^N$, 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, from suitable assumptions on $g(x,u)$ and appropriate procedures and estimates, the existence of multiple solutions can be proved for small perturbations.