Positive solutions for a class of supercritical quasilinear Schrödinger equations

This paper deals with a class of supercritical quasilinear Schrödinger equations \begin{document}$ -\Delta u+V(x)u+\kappa\Delta(\sqrt{1+{u}^{2}})\frac{u}{2\sqrt{1+{u}^{2}}} = \lambda f(u), \; x\in \mathbb{R}^{N}, $\end{document} where $ \kappa\geq2, \; N\geq3, \; \lambda > 0. $ We suppose that the nonlinearity $ f(t):\mathbb{R}\rightarrow \mathbb{R} $ is continuous and only superlinear in a neighbourhood of $ t = 0. $ By using a change of variable and the variational methods, we obtain the existence of positive solutions for the above problem.