The aim of this paper is investigating the existence of solutions of the semilinear elliptic problem \[\left\{\begin{array}{ll} \displaystyle{-\Delta u\ =\ p(x, u) + \varepsilon g(x, u)} & \mbox{ in } \Omega,\\ \displaystyle{u=0} & \mbox{ on } \partial\Omega,\\ \end{array}\right.\] where $\Omega$ is an open bounded domain of $\R^N$, $\varepsilon\in\R$, $p$ is subcritical and asymptotically linear at infinity and $g$ is just a continuous function. Even when this problem has not a variational structure on $H^1_0(\Omega)$, suitable procedures and estimates allow us to prove that the number of distinct crtitical levels of the functional associated to the unperturbed problem is ``stable'' under small perturbations, in particular obtaining multiplicity results if $p$ is odd, both in the non-resonant and in the resonant case.
Perturbed asymptotically linear problems
CANDELA, Anna Maria;SALVATORE, Addolorata
2014-01-01
Abstract
The aim of this paper is investigating the existence of solutions of the semilinear elliptic problem \[\left\{\begin{array}{ll} \displaystyle{-\Delta u\ =\ p(x, u) + \varepsilon g(x, u)} & \mbox{ in } \Omega,\\ \displaystyle{u=0} & \mbox{ on } \partial\Omega,\\ \end{array}\right.\] where $\Omega$ is an open bounded domain of $\R^N$, $\varepsilon\in\R$, $p$ is subcritical and asymptotically linear at infinity and $g$ is just a continuous function. Even when this problem has not a variational structure on $H^1_0(\Omega)$, suitable procedures and estimates allow us to prove that the number of distinct crtitical levels of the functional associated to the unperturbed problem is ``stable'' under small perturbations, in particular obtaining multiplicity results if $p$ is odd, both in the non-resonant and in the resonant case.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
