Multiple ordered solutions for a class of quasilinear problem with oscillating nonlinearity,Journal of Fixed Point Theory and Applications

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Multiple ordered solutions for a class of quasilinear problem with oscillating nonlinearity
Journal of Fixed Point Theory and Applications ( IF 1.8 ) Pub Date : 2024-02-13 , DOI: 10.1007/s11784-023-01096-2
Gelson C. G. dos Santos , Julio Roberto S. Silva

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.

更新日期：2024-02-13

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