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 \(C(\overline{\Omega })\)-norm) for the following class of problems:

$$\begin{aligned} \left\{ \begin{aligned} -&\Delta u - \kappa \Delta (u^{2}) u +\mu |u|^{q-2}u = \lambda f(u)+h(u) \ \ \text{ in } \ \ \Omega , \\ u&=0 \ \ \text{ on } \ \ \partial \Omega , \end{aligned} \right. \end{aligned}$$

where \(\Omega \) is a bounded smooth domain of \(\mathbb {R}^N\) \((N\ge 1), \kappa ,\mu ,\lambda > 0,q\ge 1\) are parameters, the nonlinearity \(f: \mathbb {R}\rightarrow \mathbb {R}\) is a continuous function that can change sign and satisfies an area condition and \(h: \mathbb {R}\rightarrow \mathbb {R}\) is a general nonlinearity.