Solutions with sign information for nonlinear Robin problems with no growth restriction on reaction

Abstract

We consider a parametric nonlinear Robin problem driven by a nonhomogeneous differential operator. The reaction is a Carathéodory function which is only locally defined (that is, the hypotheses concern only its behaviour near zero). The conditions on the reaction are minimal. Using variational tools together with truncation, perturbation and comparison techniques and critical groups, we show that for all small values of the parameter $\lambda >0$, the problem has at least three nontrivial smooth solutions, two of constant sign and the third nodal.

COMMUNICATED BY:

• Alexander Pankov

Keywords:

• Nonlinear regularity theory
• nonlinear maximum principle
• extremal constant sign solutions
• nodal solutions
• critical groups

2010 Mathematics Subject Classifications:

• 35J20
• 35J60
• 58E05

Disclosure statement

No potential conflict of interest was reported by the authors.