On the Existence of Positive Solutions for Periodic Parabolic Sublinear Problems

Similarly, let L p T be the Banach space ofT-periodic functions f such that f |Ω×(0,T) ∈ Lp(Ω× (0,T)), equipped with the norm ‖ f ‖Lp T := ‖ f |Ω×(0,T)‖Lp(Ω×(0,T)). Finally, let CT be the space of continuous and T-periodic functions onΩ×R provided with the L∞-norm. For the whole paper, we fix v,s ∈ (1,∞] such that N/2v + 1/s < 1, s > 2. Let {ai j} and {bj}, 1 ≤ i, j ≤ N , be two families of functions satisfying ai j ,b j ∈ LT and ai j = aj,i. Assume that ∑ ai j(x, t)ξiξ j ≥ α0|ξ|2 for some α0 > 0 and all (x, t) ∈ Ω× R, ξ ∈ RN . Let A be the N ×N matrix whose i, j entry is ai, j , let b = (b1, . . . ,bN ), let 0≤ c0 ∈ Ls(Lv), and let L be the parabolic operator given by Lu= ut −div(A∇u) + 〈b,∇u〉+ c0u. (1.2)