Standing waves for quasilinear Schrödinger equations with indefinite potentials

We consider quasilinear Schrödinger equations in RN of the form−Δu+V(x)u−uΔ(u2)=g(u), where g(u) is 4-superlinear. Unlike all known results in the literature, the Schrödinger operator −Δ+V is allowed to be indefinite, hence the variational functional does not satisfy the mountain pass geometry. By a local linking argument and Morse theory, we obtain a nontrivial solution for the problem. In case that g is odd, we get an unbounded sequence of solutions.