Multiple solutions for perturbed quasilinear elliptic problems - Bartolo, Rossella; Candela, Anna Maria; Salvatore, Addolorata (Topological Methods in Nonlinear Analysis, 2023)
We investigate the existence of multiple solutionsfor 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 $g(x,u)$ andappropriate procedures and estimates,the existence of multiple solutions can be proved for small perturbations.