PERIODIC PARABOLIC PROBLEMS WITH NONLINEARITIES INDEFINITE IN SIGN

Let Ω ⊂ R be a smooth bounded domain. We give sufficient conditions (which are also necessary in many cases) on two nonnegative functions a, b that are possibly discontinuous and unbounded for the existence of nonnegative solutions for semilinear Dirichlet periodic parabolic problems of the form Lu = λa (x, t) u − b (x, t) u in Ω × R, where 0 < p, q < 1 and λ > 0. In some cases we also show the existence of solutions uλ in the interior of the positive cone and that uλ can be chosen such that λ → uλ is differentiable and increasing. A uniqueness theorem is also given in the case p ≤ q. All results remain valid for the corresponding elliptic problems.

Let {a ij }, {b j }, 1 ≤ i, j ≤ N , be two families of T -periodic functions satisfying a ij ∈ C 0,1 Ω × R , a ij = a ji and b j ∈ L ∞ T , and assume that a ij (x, t) ξ i ξ j ≥ α 0 |ξ| 2 for some α 0 > 0 and all (x, t) ∈ Ω×R, ξ ∈ R N .Let A be the N×N matrix whose i, j entry is a ij , let b = (b 1 , . . ., b N ), let 0 ≤ c 0 ∈ L r T , r > N + 2, and let L be the parabolic operator given by Lu = u t − div (A∇u) + b, ∇u + c 0 u.
The existence of positive solutions for periodic parabolic problems of the form has been widely studied (see e.g.[15] and the references therein).For applications we refer to [15], [5].In [11] and [13], 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 [12], 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 inf 0<σ≤ξ (g (x, t, σ) /σ) belongs to L r T for some r > (N + 2) /2.However, in many cases of interest neither of the above conditions hold.A typical example of this situation is the problem where a, b are two nonnegative functions, 0 < p, q < 1 and λ > 0.
Our aim in this paper is to study (1.3),where a, b belong to L r T with r > N + 2. Concerning this problem, we will show existence of nonnegative solutions for all λ > 0 under weak conditions on a and b (see Theorem 3.2 and also Remark 3.3), using iterative and fixed point methods combined with some facts about linear problems with weight.Also, under different assumptions on a and b, we will prove existence of solutions in the interior of the positive cone of C 1+θ,(1+θ)/2 T for λ large enough and, for p ≤ q, that u can be chosen such that λ → u λ is differentiable and increasing (see Theorem 3.4).These last results will follow from a sub and supersolution approach together with the implicit function theorem.Finally, a uniqueness theorem for the solutions in the interior of the positive cone is given in Theorem 3.5 for the case p ≤ q.
To avoid unnecessary complexity, we restrict ourselves to (1.3), but one can see that most of the results are still valid for increasing nonlinearities that behave like u p near the origin and infinity.We mention also that as a consequence of our proofs all results remain true for the analogous elliptic problem.
In order to relate our results to others in the literature, let us mention that similar elliptic problems have been studied for example in [20] for L = −∆ and a, b ∈ C Ω using a variational approach, and recently for a, b ∈ C 1 (Ω) and allowing these functions to have a singularity in the boundary in [14, Section 3] (see also the references therein).The particular case Lu = mu p in Ω, with 0 < p < 1 and m changing sign, was treated in detail in [2] for L = −∆ and m ∈ C θ Ω , θ ∈ (0, 1), using sub and supersolutions to construct nonnegative solutions, and there is also a result for the associated parabolic initial boundary value problem there.For the one dimensional elliptic problem (1.3), a precise description of the solution set when p > q and a = b ≡ 1 was given in [8] for the operator On the other hand, in the periodic parabolic case, a similar problem to (1.3) with p = q was considered extensively in [16] for the heat operator with Neumann boundary condition, and it is also shown there that the nonnegative solutions are not necessarily unique.Existence and multiplicity of changing sign solutions for similar periodic parabolic problems were also considered in [17].
We finally mention that related elliptic superlinear problems have been widely studied too, see e.g.[1] and references therein.

Preliminaries and auxiliary results
We start collecting some known facts about the (weak) solution operator (denoted by L −1 ) and strong solutions for problem (1.1).
(iii) If µ m (λ) > 0 then for all h ∈ L r T the T -periodic problem (2.1) r,T , which is positive if h > 0, and the solution operator h → u for this problem is continuous from L r T into C T (cf.[11, Lemma 2.9]).In particular, these conclusions apply for all λ ≥ 0 if λ 1 (m) does not exist and for 0 ≤ λ < λ 1 (m) when λ 1 (m) exists.
(iv) The following comparison principle holds: if The following lemma states a maximum principle for noncylindrical domains.
Then u ≥ 0 in D.
Remark 2.4.In Lemma 2.3, the regularity assumptions on the coefficients of L can be weakened.In fact, it is clear from the proof that it suffices that they hold in each compact subset of D.

The main results
Lemma 3.1.Let b, f ∈ L r T , r > N + 2, b ≥ 0.Then, there exists a unique u ∈ W 2,1 r,T solution of Moreover, the solution operator S : C T → C T is compact and increasing.In particular, f > 0 implies Sf > 0.
Proof: Let v ∈ C T .From Remark 2.1 (ii) we have that there exists a unique solution u ∈ W 2,1 r,T of the Dirichlet periodic problem We note that the solution operator S f : C T → C T is compact.Indeed, Lebesgue's dominated convergence theorem gives that the map v → |v| q−1 v is continuous from C T into L s T for all s ≥ 1, and therefore v → −b |v| q−1 v + f is continuous from C T into L l T for l > N + 2. Thus, the compactness follows from Remark 2.1 i).
For R > 0, let B CT R be the closed ball in C T with center at 0 and radius R. We claim that S f : Hence, |u| ≤ c 1 + c 2 R q where c 1 , c 2 do not depend on u and the claim follows.So, the Schauder fixed point theorem (e.g.[9, Corollary 11.2]) gives a solution for (3.1).
Suppose now there exist two solutions u, w ∈ W 2,1 r,T for (3.1).Then Since |w| q−1 w − |u| q−1 u / (w − u) is always positive when u = w, the maximum principle in Lemma 2.3 implies u = w.
To end the lemma, we note that the compactness of S follows reasoning as in the beginning of the proof, and the fact that S is increasing follows by Lemma 2.3 in a similar way as above.Theorem 3.2.Let 0 < p, q < 1, and let 0 ≤ a, b ∈ L r T , r > N + 2, such that the following condition holds (H1) There exists an open set B := Ω 0 × (t 0 , t 1 ) ⊂ Ω × R such that a (x, t) > 0 in a subset of B of positive measure and b ≡ 0 in B.
Proof: Let 0 ≤ v ∈ C T and λ > 0. By the above lemma there exists a unique positive solution u ∈ W 2,1 r,T of (3.1) with f := λav p .Let S f be the operator defined by S f v = S (λav p ) where S is the solution operator of Lemma 3.1.Then S f is compact and nondecreasing.Note that if v ≡ c for some constant c large enough, then S f v ≤ v. Indeed, let u = S f v. From (3.1) it follows that Lu ≤ λav p and so u ∞ ≤ Kc p for some K independent of c.
Take v ≡ c ≫ 0, consider the nonincreasing sequence and let u ∞ ≥ 0 be its limit.We have where u j = S j f v. Now, by Lebesgue's theorem the right side of (3.3) converges in L r T , thus {u j+1 } ∞ j=1 converges in W 2,1 r,T (and so also in r,T and u ∞ = 0 on ∂Ω × R. Going to the limit in (3.3) we find that It remains to see that u ∞ = 0. We proceed by contradiction.Suppose that u ∞ = 0. Since u j converges to u ∞ in C T , there exists j 0 such that u j ≤ 1 for j ≥ j 0 .Observe also that (3.4) u k (x, t) > 0 for (x, t) ∈ B, k ∈ N ∪ {0} .
Let λ > λ 0 , let u λ be the solution of (1.3) found above, and let m λ := λpau p−1 λ − qbu q−1 λ , m λ := λau p−1 λ − bu q−1 λ .Note that (H2) and (H3) imply that m λ and m λ belong to L s T for some s > N + 2. We claim that the implicit function theorem can be applied in the point (λ, u λ ).Indeed, to see this it suffices to show that for a given g ∈ L r T there is a unique solution h ∈ W 2,1 r,T of the problem (3.9) and that the solution operator S λ for this problem is continuous.Consider first the case when λ 1 (m λ ) exists.From (1.3) it follows that λ 1 ( m λ ) = 1.Since m λ ≥ m λ /q, recalling the comparison principle in Remark 2.2 (iv) we obtain and so µ m λ (1) > 0. If λ 1 (m λ ) does not exist the same conclusion holds.Hence, by the results in Remark 2.2 (iii) and Remark 2.1 (ii) in both cases S λ is well defined and continuous on W 2,1 r,T .Let (α, β) be a maximal interval in which λ → u λ is a C 1 map into W 2,1 r,T ∩ P • .Observe that λ → u λ is increasing.Indeed, differentiating (1.3) with respect to λ gives If λ 1 (m λ ) exists, Remark 2.2 (iii) and (3.10) imply ∂u ∂λ > 0. If λ 1 (m λ ) does not exist, then µ m λ is positive everywhere, thus µ m λ (1) > 0 and so, again by Remark 2.2 (iii) we get ∂u ∂λ > 0 also in this case.Now, suppose β < ∞, and let u j ∈ P • be the solutions of (1.3) corresponding to some sequence λ j → β − .Recalling Remark 2.1 (i), a standard compactness argument gives some u β solution of (1.3) for λ = β.
Moreover, since λ → u λ is increasing we have u β ∈ P • .But then reasoning as above we can apply the implicit function theorem in the point (β, u β ), contradicting the maximality of (α, β).
Concerning the matter of uniqueness or multiplicity of positive solutions, there are examples of different nonnegative solutions for similar problems to (1.3) with p = q both in the elliptic and the periodic parabolic case (cf.[2], [16]), so we cannot expect uniqueness here.Moreover, when p > q, a = b ≡ 1, L = −∆ p and N = 1, it is shown in [8] that for some λ there are exactly two solutions in the interior of the positive cone for (1.3), and therefore we cannot expect a general uniqueness theorem for the case p > q neither.However, when p ≤ q the solution in the interior of the positive cone for (1.3) (if such a solution exists) turns out to be unique as the following theorem shows.Its proof is inspired in the proof of Proposition 2.2 in [6] and uses a change of variable introduced (for the elliptic case and b = 0) by L. Nirenberg (see [3, Appendix II, Method IV]).Theorem 3.5.Let 0 < p ≤ q < 1, λ > 0, and let 0 ≤ a, b ∈ L r T , r > N + 2. Then there exists at most one solution u ∈ W 2,1 r,T ∩ P • of (1.3).