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 q, r 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).