Nonlinear flux “concave–convex” problems: a fibering method approach

Abstract

This paper studies the nonlinear flux problem:

where \(\varDelta _p\) stands for the p-Laplacian operator, \(\varOmega \subset {\mathbb {R}}^N\) is a bounded smooth domain, \(\lambda\) is a positive parameter and \(\nu\) stands for the outer unit normal at \(\partial \varOmega\). The exponents qr are assumed to vary in the concave convex regime \(1< q< p < r\) while \(1< p < N\) and r is subcritical \(r < p^*\). Our objective here is showing the existence, for every \(0< \lambda < {{\bar{\lambda }}}\), of two different sets of infinitely many solutions of (P). The energy functional associated to the problem exhibits a different sign on each of these sets. The analysis of positive energy solutions involves the so-called fibering method (Drábek and Pohozaev in Proc R Soc Edinb Sect A 127(4):703–726, 1997). Our results have been inspired by similar ones in García-Azorero et al. (J Differ Equ 198(1):91–128, 2004), García-Azorero and Peral (Trans Am Math Soc 323(2):877–895, 1991) and El Hamidi (Commun Pure Appl Anal 3(2):253–265, 2004). This work can be considered as a natural continuation of Sabina de Lis (Differ Equ Appl 3(4):469–486, 2011), Sabina de Lis and Segura de León (Adv Nonlinear Stud 15(1):61–90, 2015) and Sabina de Lis and Segura de León (Nonlinear Anal 113:283–297, 2015). The main achievement of the latter of these works consisted in showing a global existence result of positive solutions to (P).

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Acknowledgements

The author is very grateful to the anonymous referee for his/her careful revision of the original manuscript. Supported by Dirección General de Investigación under Grant MTM2014-52822-P.

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Correspondence to José C. Sabina de Lis.

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Communicated by Julio Rossi.

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Sabina de Lis, J.C. Nonlinear flux “concave–convex” problems: a fibering method approach. Adv. Oper. Theory (2020). https://doi.org/10.1007/s43036-020-00092-4

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Keywords

  • Variational methods
  • Minimax methods
  • Degenerate diffusion

Mathematics Subject Classification

  • 35J20
  • 35J70