Two solutions for a fourth order nonlocal problem with indefinite potentials (Q2313939)

In this paper the authors give sufficient conditions for the existence of a positive solution and of a second (possibly sign changing) solution for nonlocal problems \[ \begin{cases} & \Delta^2 u- m\left(\int_\Omega |\nabla u|^2\ dx\right)\Delta u = \lambda a(x)|u|^{q-2}u+b(x) |u|^{p-2}u, \text{ in }\Omega\\ & u = \Delta u =0, \text{ on } \partial\Omega. \end{cases}\] Here \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\) with \(N\ge 5\) and the exponents on the right hand side satisfy \[ 1< q < 2 < p \le \frac{2N}{N-4}. \] The method of proof is variational, and special care has to be taken in the critical case \(p=\frac{2N}{N-4}.\)