A Landesman–Lazer type result for periodic parabolic problems on RN at resonance. (September 2015) - Ćwiszewski, Aleksander; Łukasiak, Renata
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.