Three solutions for a perturbed Navier problem

Abstract

In this paper we prove the existence of at least three distinct solutions to the following perturbed Navier problem:

$$\left\{\begin{array}{ll}\Delta (|{\Delta u}|^{p-2}\Delta u) = f(x,u) + \lambda g(x,u) \quad{\rm in}\,\,\,\Omega \\ u=\Delta u = 0 \qquad\qquad\qquad\qquad\qquad\quad{\rm on}\,\,\, \partial \Omega,\end{array}\right.$$

where ${{\Omega \subset \mathbb {R}^N}}$ is an open bounded set with smooth boundary ${\partial \Omega}$ and ${\lambda \in \mathbb {R}}$ . Under very mild conditions on g and some assumptions on the behaviour of the potential of f at 0 and +∞, our result assures the existence of at least three distinct solutions to the above problem for λ small enough. Moreover such solutions belong to a ball of the space ${W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)}$ centered in the origin and with radius not dependent on λ.