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

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Abstract

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.

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Funding

The first author was partially supported by CNPq/Brazil Proc. No 306709-2022-8 and by FAPDF/Brazil.

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  1. Faculdade de Matemática, Universidade Federal do Pará, Belém, PA, 66075-110, Brazil

    Gelson C. G. dos Santos

  2. Universidade Federal do Pará Campus Universitário de Cametá, Cametá, PA, CEP: 68.400-000, Brazil

    Julio Roberto S. Silva

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  1. Gelson C. G. dos Santos
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dos Santos, G.C.G., Silva, J.R.S. Multiple ordered solutions for a class of quasilinear problem with oscillating nonlinearity. J. Fixed Point Theory Appl. 26, 7 (2024). https://doi.org/10.1007/s11784-023-01096-2

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