On the existence of positive solutions for periodicparabolic sublinear problems

Abstract

We give necessary and sufficient conditions for the existence of positive solutions for sublinear Dirichlet periodic parabolic problems Lu = g (x,t,u) in Ω × R (where Ω ⊂ R N is a smooth bounded domain) for a wide class of Carathéodory functions g : Ω × R × [0,∞) → R satisfying some integrability and positivity conditions.

Periodic parabolic problems with concave and convex nonlinearities

“…Theorems 3.3 and 3.4 in [9] give some positive solution v ∈ L ∞ T in cases (i) and (ii) respectively. Since the operator…”

“…Note that it suffices to show that there exists a supersolution u of (1.3) for λ = Λ 2 . Indeed, this is enough because Lemma 2.4 in [9] gives subsolutions of this problem of the form εΦ (Φ ∈ P…”

“…In [8] and [11], bifurcation of positive solutions for (1.2) was proved assuming that ξ → g (x, t, ξ) /ξ is nonincreasing in (0, ∞) and that g ξ (x, t, 0) belongs to L r T for some r > (N + 2) /2. On the other hand, in [9], existence results of positive solutions for (1.2) were given without monotonicity conditions on g and allowing g ξ (x, t, 0) = +∞, but assuming that sup 0<ξ≤σ…”

“…On the other hand, for a non-selfadjoint operator, some existence results have been proved recently for 0 ≤ a, b ∈ C 1 (Ω) and allowing these functions to have a singularity in the boundary in [12], Section 3, and also in [4] assuming that a ≡ 1 and b ∈ C α Ω , α ∈ (0, 1). In the parabolic case, when b ≤ 0 (logistic-type equation), existence of positive periodic solutions was shown in [9], but no results seem to be known for (1.3) when b > 0 in a set of positive measure. We refer to [15], [14], [9], [10] and its references for similar periodic parabolic problems.…”

“…In the parabolic case, when b ≤ 0 (logistic-type equation), existence of positive periodic solutions was shown in [9], but no results seem to be known for (1.3) when b > 0 in a set of positive measure. We refer to [15], [14], [9], [10] and its references for similar periodic parabolic problems.…”