Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 12, pp. 1-20.

Title: Positive and nodal solutions for nonlinear nonhomogeneous parametric Neumann problems

Authors: Nikolaos S. Papageorgiou (National Technical Univ., Athens, Greece)
         Calogero Vetro (Univ. of Palermo, Palermo, Italy)
         Francesca Vetro (Ton Duc Thang Univ., Ho Chi Minh City, Vietnam)

Abstract:
 We consider a parametric Neumann problem driven by a nonlinear
 nonhomogeneous differential operator plus an indefinite potential term.
 The reaction term is superlinear but does not satisfy the
 Ambrosetti-Rabinowitz condition.
 First we prove a bifurcation-type result describing in a precise way the
 dependence of the set of positive solutions on the parameter $\lambda>0$.
 We also show the existence of a smallest positive solution. Similar results
 hold for the negative solutions and in this case we have a biggest
 negative solution. Finally using the extremal constant sign solutions
 we produce a smooth nodal solution. 
 
Submitted February 11, 2019.  Published January 24, 2020.
Math Subject Classifications: 35J20, 35J60, 58E05.
Key Words: Nonlinear nonhomogeneous differential operator; 
           nonlinear regularity theory; nonlinear maximum principle; 
           strong comparison; bifurcation-type theorem;  nodal solution; 
           critical group.