Standing waves for quasilinear Schr\"{o}inger equations with indefinite potentials

We consider quasilinear Schr"{o}dinger equations in ${\mathbb{R}}^N$ of the form% $$-\mathrm{\Delta }u+V(x)u-u\mathrm{\Delta }(u^2)=g(u)\text{,}$$ where $g(u)$ is $4$-superlinear. Unlike all known results in the literature, the Schr"{o}dinger operator $-\mathrm{\Delta }+V$ 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.