Indefinite Fourth Order Elliptic Equations with Decay or Integrable Weights

We consider fourth order elliptic equations in \(\mathbb{R}^N\) of the form \[\Delta^2u-\Delta u+V(x)u-\frac{1}{2}u\Delta(u^2)=f(x,u),\] where the operator \(\Delta^2-\Delta+V\) is allowed to be indefinite. We study the case that the potential \(V\) is bounded so that the working space is \(H^2(\mathbb{R}^N)\), which can not be compactly embedded into Lebesgue spaces. To overcome this difficulty, the nonlinearity considered here satisfies decay or integrable weights. By local linking theorem, we obtain a nontrivial solution for the problem.