Nonlinear flux "concave-convex" problems: a fibering method approach

This paper studies the nonlinear flux problem: where AI, stands for the p -Laplacian operator, 12 c RN is a bounded smooth domain, 2 is a positive parameter and v stands for the outer unit normal at a Q. 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 <2 <2, 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 (Drabek and Pohozaev in Proc R Soc Edinb Sect A 127(4):703-726, 1997). Our results have been inspired by similar ones in Garcia-Azorero et al. (J Differ Equ 198(1):91-128, 2004), Garcia-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 Leon (Adv Nonlinear Stud 15(1):61-90, 2015) and Sabina de Lis and Segura de Leon (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).