Electron. J. Differential Equations, Vol. 2020 (2020), No. 12, pp. 1-20.

Positive and nodal solutions for nonlinear nonhomogeneous parametric Neumann problems

Nikolaos S. Papageorgiou, Calogero Vetro, Francesca Vetro

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.

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  Nikolaos S. Papageorgiou
National Technical University
Department of Mathematics, Zografou campus
15780, Athens, Greece
email: npapg@math.ntua.gr
Calogero Vetro
University of Palermo
Department of Mathematics and Computer Science
Via Archirafi 34, 90123, Palermo, Italy
email: calogero.vetro@unipa.it
Francesca Vetro
Nonlinear Analysis Research Group
Faculty of Mathematics and Statistics
Ton Duc Thang University
Ho Chi Minh City, Vietnam
email: francescavetro@tdtu.edu.vn

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