Two solutions for a fourth order nonlocal problem with indefinite potentials

Abstract

We study the nonlocal equation $$\begin{aligned} \Delta ^{2}u-m\left( \displaystyle \int _{\Omega }|\nabla u|^{2} dx \right) \Delta u = \lambda a(x) |u|^{q-2}u+ b(x)|u|^{p-2}u, \, \text{ in } \Omega , \end{aligned}$$ subject to the boundary condition $$u=\Delta u=0$$ on $$\partial \Omega $$ . For m continuous and positive we obtain a nonnegative solution if $$1<q<2<p \le 2N/(N-4)$$ and $$\lambda >0$$ small. If the affine case $$m(t)=\alpha +\beta t$$ , we obtain a second solution if $$4<p<2N/(N-4)$$ and $$N \in \{5,6,7\}$$ . In the proofs we apply variational methods.