Quasilinear Schrödinger equations with concave and convex nonlinearities

In this paper, we consider the following quasilinear Schrödinger equation \begin{align*} -Δu-uΔ(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 $1<q<2<s<+\mathrm{\infty }$. Unlike most results in the literature, the exponent $s$ here is allowed to be supercritical $s>2\cdot 2^{\ast }$. 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.