Nodal solutions for nonlinear nonhomogeneous Robin problems

Abstract

We consider the nonlinear Robin problem driven by a nonhomogeneous differential operator plus an indefinite potential. The reaction term is a Carathéodory function satisfying certain conditions only near zero. Using suitable truncation, comparison, and cut-off techniques, we show that the problem has a sequence of nodal solutions converging to zero in the C 1 (Ω)-norm.

Constant Sign and Nodal Solutions for Nonlinear Robin Equations With Locally Defined Source Term

“…The work of Wang [20] was extended by Li-Wang [7] to semilinear Schrödinger equations. Extensions of these results were obtained by Papageorgiou-Vetro-Vetro [18] (semilinear Robin problems), Papageorgiou-Rǎdulescu-Repovš [15] (nonlinear Robin problems) and Papageorgiou-Rǎdulescu-Repovš [14] (Dirichlet (p, 2)-equations). In all the aforementioned works the source term is symmetric near zero and this leads to an application of a version of the symmetric mountain pass theorem, which generates the desired sequence of distinct nodal solutions.…”

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

“…Also, we have * and so u λ * = 0, which contradicts our hypothesis that u λ * = 0. Therefore u λ * = 0 and by passing to the limit as n → +∞ in (15) and using (19), we infer that u λ * ∈ S + λ ⊆ D + and u λ * = inf S + λ .…”

“…If in (15) we choose h = u n ∈ W 1,p (Ω) and we use ( 13), ( 16) and hypothesis H(f) (i) we infer that (17) {u n } n≥1 ⊆ W 1,p (Ω) is bounded.…”

“…No such condition is used in this work. We also mention the recent work of Papageorgiou-Rǎdulescu-Repovš [15], who impose a symmetry condition on f (z, •) and produce a whole sequence of distinct nodal solutions converging to zero in C 1 (Ω).…”