A Landesman–Lazer type result for periodic parabolic problems on RN at resonance. (September 2015)

• Record Type: Journal Article
• Authors: Ćwiszewski, Aleksander; Łukasiak, Renata
• Is Part Of: Nonlinear analysis. Volume 125(2015)
• Page Start: 608
• Page End: 625
• Publication Date: 2015-09
• Subjects: Parabolic partial differential equation -- Periodic solution -- Resonance -- Fixed point index -- Translation operator; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.06.001
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We are concerned with T -periodic solutions of nonautonomous parabolic problem of the form u t = Δ u + V ( x ) u + f ( t, x, u ), t > 0, x ∈ R N, with V ∈ L ∞ ( R N ) + L p ( R N ), where 2 < p < ∞ if N = 1, 2 and N ≤ p < ∞ for N ≥ 3 and T -periodic continuous perturbation f : R N × R → R . The so-called resonant case is considered, i.e. when N : = Ker ( Δ + V ) ≠ { 0 } and f is bounded. We derive a formula for the fixed point index of the associated translation along trajectories operator in terms of the Brouwer topological degree of the time averaging of the mapping f ˆ : N → N, f ˆ being the restriction of f to N . By use of the formula and continuation techniques we show that Landesman–Lazer type conditions imply the existence of T -periodic solutions.