  In this paper, we consider the following quasilinear Schr\"{o}dinger equation
\begin{align*} -\Delta u-u\Delta(u^{2})=k(x)\left\vert u\right\vert
^{q-2}u-h(x)\left\vert u\right\vert ^{s-2}u\text{, }u\in
D^{1,2}(\mathbb{R}^{N})\text{,} \end{align*} where $1<q<2<s<+\infty$. Unlike
most results in the literature, the exponent $s$ here is allowed to be
supercritical $s>2\cdot2^{\ast}$. By taking advantage of geometric properties
of a nonlinear transformation $f$ and a variant of Clark's theorem, we get a
sequence of solutions with negative energy in a space smaller than
$D^{1,2}(\mathbb{R}^{N})$. Nonnegative solution at negative energy level is
also obtained.
