Multiple ordered solutions for a class of quasilinear problem with oscillating nonlinearity

In this paper, we use truncation argument combined with method of minimization, argument of comparison, topological degree arguments and sub-supersolutions method to show existence of multiple positive solutions (which are ordered in the \(C(\overline{\Omega })\) $C(\overline{\mathrm{\Omega }})$ -norm) for the following class of problems: \(\begin{aligned} \left\{ \begin{aligned} -&\Delta u - \kappa \Delta (u^{2}) u +\mu |u|^{q-2}u = \lambda f(u)+h(u) \ \ \text{ in } \ \ \Omega , \\ u&=0 \ \ \text{ on } \ \ \partial \Omega , \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill -& \mathrm{\Delta }u-\kappa \mathrm{\Delta }(u^2)u+\mu {|u|}^{q-2}u=\lambda f(u)+h(u)\text{in}\mathrm{\Omega },\hfill \\ \hfill u& =0\text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded smooth domain of \(\mathbb {R}^N\) ${\mathbb{R}}^N$ \((N\ge 1), \kappa ,\mu ,\lambda > 0,q\ge 1\) $(N\ge 1),\kappa ,\mu ,\lambda >0,q\ge 1$ are parameters, the nonlinearity \(f: \mathbb {R}\rightarrow \mathbb {R}\) $f:\mathbb{R}\to \mathbb{R}$ is a continuous function that can change sign and satisfies an area condition and \(h: \mathbb {R}\rightarrow \mathbb {R}\) $h:\mathbb{R}\to \mathbb{R}$ is a general nonlinearity.