Nonlinear, Nonhomogeneous Periodic Problems with no Growth Control on the Reaction

We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator, which includes as a particular case the scalar p-Laplacian. We assume that the reaction is a Carathéodory function which admits time-dependent zeros of constant sign. No growth control near ±∞ is imposed on the reaction. Using variational methods coupled with suitable truncation and comparison techniques, we prove two multiplicity theorems providing sign information for all the solutions.


Introduction
In this paper, we study the following nonlinear periodic problem: (1.1) curvature differential operator. The reaction f (t, ζ ) is a Carathéodory function (i.e. for all ζ ∈ R, the function t −→ f (t, ζ ) is measurable, and for almost all t ∈ T , the function ζ −→ f (t, ζ ) is continuous) which has cosign, t-dependent zeros. Our aim is to prove multiplicity theorems for problem (1.1), providing precise sign information for all the solutions.
In fact, our conditions on the reaction f (t, ζ ) are simple and easy to verify and incorporate into our framework several interesting applied cases. Essentially, we require that the reaction f (t, ·) exhibits a kind of oscillatory behaviour near zero. For example, consider the following semilinear periodic problem: with α, β, γ > 0 such that β 2 − 4αγ > 0. For this problem, the reaction is autonomous (t-independent) and has the form Since β± √ β 2 −4αγ 2γ > 0, there exist 0 < ζ 0 < ζ 1 such that f (ζ 0 ) = f (ζ 1 ) = 0. Then according to Proposition 3.4, this problem has a positive solution. This equation is a homogeneous version of a problem studied by Cronin-Scanlon [8] in the context of a biomathematical model of aneurysm. In fact, we can add in the reaction a suitable perturbation h(ζ ) with no growth restriction, provided that it has suitable oscillatory behaviour near zero.
Our framework also incorporates logistic equations of the following form: with q > 2. In this case, f (ζ ) = ζ − ζ q−1 = ζ(1 − ζ q−2 ), ζ > 0, and we infer that the problem has a positive solution. Of course we can have a reaction of the form f (ζ ) = ζ − |ζ | q−2 ζ , ζ ∈ R, and then we can guarantee also negative solutions (see Proposition 3.4). We may include harvesting, that is Usually the harvesting is proportional to the population, that is h(ζ ) = cζ , c > 0. Then If c ∈ (0, 1), then we are back to the previous situation. In fact for such problems, the function ζ −→ f (ζ ) ζ is strictly decreasing on (0, +∞) and so according to , the positive solution is unique.
Other possibility is a reaction of the form with q > 2, which arises in chemotaxis models. The reaction f (ζ ) = |ζ | τ −2 ζ − |ζ | q−2 ζ, ζ ∈ R, with τ < p < q, leads to a logistic-type equation of subdiffusive type and fits in the framework of Theorem 3.11. So, the corresponding equation driven by the scalar p-Laplacian has at least three solutions, two of constant sign and the third nodal.
Thus, we see that our setting is general and rather natural in the context of many applied problems.
In this paper, we prove two "three-solution theorems," in which we produce a positive, a negative and a nodal (sign changing) solutions.
The assumption that f (t, ·) has zeros implies that we do not need to impose any growth control near ±∞ for the function (t, ·). Our approach is variational based on the critical point theory, coupled with suitable truncation and comparison techniques.
In the next section, for the convenience of the reader, we present the main mathematical tools which we will use in this work.

Mathematical Background -Hypotheses
Let X be a Banach space and let X * be its topological dual. By ·, · , we denote the duality brackets for the pair (X, X * ). We say that ϕ ∈ C 1 (X) satisfies the Palais-Smale condition if the following is true: admits a strongly convergent subsequence." Using this compactness-type condition, we can prove the following minimax theorem, known in the literature as the "mountain pass theorem." then c η r and c is a critical value of ϕ.
Another result from critical point theory which we will need in the sequel is the so-called second deformation theorem (see, e.g. p. 628]). Let ϕ ∈ C 1 (X) and let c ∈ R. We introduce the following sets:

Remark 2.3
In particular, Theorem 2.2 implies that ϕ a is a strong deformation retract of ϕ b \ K b ϕ . Hence, the two sets are homotopy equivalent.
In the study of problem (1.1), we will use the following two spaces: where 1 < p < +∞. Recall that the Sobolev space W 1,p (0, b) is embedded continuously (in fact compactly) in C(T ), and so the evaluations at t = 0 and t = b of u ∈ W 1,p (0, b) make sense. The Banach space C 1 (T ) is an ordered Banach space with a positive cone This cone has a nonempty interior given by int C + = u ∈ C + : u(t) > 0 for all t ∈ T .
Consider the following nonlinear eigenvalue problem: where 1 < p < +∞. A number λ ∈ R is said to be an eigenvalue of the negative periodic scalar p-Laplacian if problem (2.1) has a nontrivial solution, which is a corresponding eigenfunction. Evidently, a necessary condition for λ ∈ R to be an eigenvalue is that λ 0. We see that λ 0 = 0 is an eigenvalue and the corresponding eigenfunctions are constant functions (i.e. the corresponding eigenspace is R). Let Then λ n = Let u 0 be the L p -normalized principal (i.e. corresponding to λ 0 = 0) eigenfunction. Hence, Also, let For λ 1 > 0 (the first nonzero eigenvalue), we have the following variational characterization (see Aizicovici-Papageorgiou-Staicu [4,5]).

Example 2.6
The following functions a(·) satisfy hypotheses H (a): This map corresponds to the (p, q)-Laplace differential operator (the sum of a scalar p-Laplacian with a scalar q-Laplacian).
This map corresponds to the scalar generalized p-mean curvature operator.
In what follows, for notational economy, we write W = W 1,p (

Proposition 2.8 If hypotheses H (a) hold and u
then u 0 ∈ C 1 (T ) and it is also a local W -minimizer of σ 0 , i.e. there exists 1 > 0, such that Throughout this paper, by · , we denote the norm of the Sobolev space W = W 1,p we denote the weak convergence in any Banach space. If ζ ∈ R, then we set ζ + = max{ζ, 0} and ζ − = max{−ζ, 0}.
We know that u + , u − ∈ W and u = u + − u − , |u| = u + + u − . By | · | 1 we denote the Lebesgue measure on R and if h : T × R −→ R is a measurable function (for example, a Carathéodory function), then we set

Three Solution Theorems
In this section, we prove two multiplicity theorems for problem (1.1) providing sign information for all the solutions.
To produce the constant sign solutions, we will need the following hypotheses on the reaction f : (iv) there exists ξ * > 0, such that

Remark 3.1
Hypotheses H (f ) 1 (ii) and (iii) imply that for almost all t ∈ T , f (t, ·) has t-dependent zeros of constant sign. The presence of these zeros frees f (t, ·) from any growth restrictions near ±∞. Note that we do not impose any control on the growth of We start by showing that the nontrivial constant sign solutions of Eq. 1.1 have L ∞ norms which are bounded away from zero.

Proposition 3.2 If hypotheses H (a)
and Proof Since by hypothesis u ∈ C + \ {0} is a solution of Eq. 1.1, we have (since 0 u(t) < δ 0 for all t ∈ T ; see hypothesis H (f ) 1 (iii)). This contradicts hypothesis Next, we establish the existence of nontrivial solutions of constant sign.

Proposition 3.3 If hypotheses H (a)
and H (f ) hold, then problem (1.1) has at least one nontrivial positive solution u 0 ∈ int C + and at least one nontrivial negative solution v 0 ∈ −int C + .
Proof First, we produce the nontrivial positive solution. To this end, we consider the following truncation-perturbation of the reaction f : This is a Carathéodory function. Let

s) ds
and consider the C 1 -functional ϕ + : W −→ R, defined by It is clear from Eq. 2.3 and Eq. 3.2 that ϕ + is coercive. Also, using the Sobolev embedding theorem, we see that ϕ + is sequentially weakly lower semicontinuous. So, by virtue of the Weierstrass theorem, we can find u 0 ∈ W , such that Let ξ ∈ (0, δ 0 ]. Then, for For the nontrivial negative solution, we consider This is a Carathéodory function. We set

s) ds
and consider the C 1 -functional ϕ − : W −→ R, defined by Reasoning as above, via the direct method, we obtain a nontrivial negative solution v 0 ∈ −int C + .
In fact, we can show that (1.1) admits extremal nontrivial constant sign solution, i.e. there is the smallest nontrivial positive solution and biggest nontrivial negative solution. H (a) and H (f ) hold, then problem (1.1) has the smallest nontrivial positive solution u * ∈ int C + and biggest nontrivial negative solution v * ∈ −int C + .

Proposition 3.4 If hypotheses
Proof First, we show the existence of the smallest nontrivial positive solution. Let ξ ∈ (0, δ 0 ] (where δ 0 > 0 is as in hypothesis H (f ) 1 (iii)) and consider the order interval To this end, we consider the following truncation-perturbation of f (t, ·): This is a Carathéodory function. Let

(ii), we have
so u w + (as before, see hypothesis H (a)(i)). Therefore, we have proved that u ∈ [ξ, w + ]. This by virtue of Eq. 3.5 and Eq. 3.7 implies that u(t)) a.e. on T , and thus u ∈ C 1 (T ) is a solution of Eq. Let Y + be the set of solutions of problem (1.1) in the order interval [ξ, w + ]. From Claim 1, we know that Y + = ∅. Let C ⊆ Y + be a chain (i.e. a nonempty totally ordered subset of Y + ). From Dunford-Schwartz [10, p.336], we know that we can find a sequence We have A(u n ) = N f (u n ) and u n ∈ [ξ, w + ] ∀n 1, (3.8) so the sequence {u n } n 1 ⊆ W is bounded. So, we may assume that Acting on Eq. 3.8 with u n − u ∈ W , passing to the limit as n → +∞ and using Eq. 3.9, we obtain lim So, if in Eq. 3.8, we pass to the limit as n → +∞ and use Eq. 3.10, we have so u ∈ Y + and u = inf C. Since C is an arbitrary chain, from the Kuratowski-Zorn lemma, we infer that Y + has a minimal element u ∈ Y + . Exploiting the monotonicity of A (see Proposition 2.3), as in Aizicovici-Papageorgiou-Staicu [3] (see Lemma 1 and Proposition 8), we show that Y + is downward directed (i.e. if u 1 , u 2 ∈ Y + , then we can find u ∈ Y + , such that u u 1 , u u 2 ). Hence, u ∈ Y + is the smallest solution of Eq. 1.1 in the order interval [ξ, w + ]. This proves Claim 2. Now suppose that {ξ n } n 1 ⊆ (0, δ 0 ] is a sequence, such that ξ n 0. By virtue of Claim 2, for every n 1, we can find the smallest solution u n ∈ C 1 (T ) of Eq. 1.1 in [ξ n , w + ]. Then, {u n } n 1 ⊆ W is bounded decreasing, and we may assume that u n w −→ u * in W and u n −→ u * in C(T ), so u * ∞ δ 0 (see Proposition 3.2) and thus u * = 0.
Similarly, for the negative solution, we choose ξ ∈ [−δ 0 , 0) and consider the order interval Then, the set Y − of nontrivial solutions of problem (1.1) in [w − , ξ] is nonempty and upward [3]). So, as above, we can find the biggest nontrivial negative solution v * ∈ −int C + of problem (1.1).
Using these extremal nontrivial constant sign solutions, we will produce a nodal (sign changing) solution. To this end, we need to restrict further the behaviour of f (t, ·) near zero. More precisely, the new hypotheses on the reaction f are the following: Carathéodory function, such that f (t, 0) = 0 for almost all t ∈ T , hypotheses H (f ) 2 (i), (ii) and (iv) are the same as the corresponding hypotheses H (f ) 1 (i), (ii), (iv) and (iii) there exist q ∈ (1, τ ) and δ 0 > 0, such that

Remark 3.5 Clearly hypothesis H (f ) 2 (iii) is more restrictive than hypothesis H (f ) 1 (iii)
and we can easily see that it implies that F (t, ζ ) c 3 |ζ | q for almost all t ∈ T , all |ζ | δ 0 with some c 3 > 0.
With these stronger hypotheses on f (t, ·), we can produce a nodal solution.
Proof Let u * ∈ int C + and v * ∈ int C + be the two extremal nontrivial constant sign solutions produced in Proposition 3.4. Using them, we introduce the following truncationperturbation of the reaction f (t, ·): This is a Carathéodory function. We set

s) ds
and consider the C 1 -functional σ : W −→ R, defined by Also, let

s) ds
and consider the C 1 -functional σ ± : W −→ R, defined by As in the proof of Proposition 3.4, we can show that The extremality of the solutions u * and v * implies that Claim. u * and v * are local minimizers of σ . Evidently, the functional σ + is coercive (see Eq. 3.11). Also, it is sequentially weakly lower semicontinuous. So, we can find u ∈ W , such that As before, hypothesis H (f ) 2 (iii) implies that hence u = 0. Since u ∈ K σ + , from Eq. 3.12, it follows that u = u * ∈ int C + . But note that Because u * ∈ int C + , it follows that u * is a local C 1 (T )-minimizer of σ . Invoking Proposition 2.8, we infer that u * is a local W -minimizer of σ . Similarly for v * using this time the functional σ − . This proves the Claim. Without any loss of generality, we may assume that σ (v * ) σ (u * ) (the analysis is similar if the opposite inequality holds). Then, as in Aizicovici-Papageorgiou-Staicu [2, Proposition 29] or Gasiński-Papageorgiou [14, proof of Theorem 3.4], we can find ∈ (0, 1) small, such that Since the functional σ is coercive (see Eq. 3.11), it satisfies the Palais-Smale condition. Indeed, let {u n } n 1 ⊆ W be such that the squence {σ (u n )} n 1 ⊆ R is bounded and σ (u n ) −→ 0 in W * . (3.14) From the coercivity of σ , it follows that {u n } n 1 ⊆ W is bounded, and so we may assume that u n w −→ u in W and u n −→ u in C(T ).
Then as before, using the convergence in Eq. 3.14 and Proposition 2.7, we conclude that hence σ satisfies the Palais-Smale condition. This fact and (3.13) permit the use of the mountain pass theorem (see Theorem 2.1). So, we can find y 0 ∈ W , such that y 0 ∈ K σ and η σ (y 0 ), so y 0 ∈ C 1 (T ) solves problem (1.1), y 0 ∈ [v * , u * ] (see Eq. 3.12), y 0 = v * and y 0 = u * (see Eq. 3.13). It remains to show that y 0 is nontrivial. We know that y 0 is a critical point of σ of mountain pass type, while hypothesis H (f ) 2 (iii) implies the presence of a concave term near the origin. Hence, the origin is a critical point of a different kind and must be different from y 0 . An easy way to establish this rigorously is to use critical groups. Since y 0 ∈ K σ is of mountain pass type, we have C 1 (σ, y 0 ) = 0 (3.15) (see Chang [7, p. 89]). On the other hand, hypothesis H (f ) 2 (iii) and Proposition 2.1 of Moroz [20] imply that C k (σ, 0) = 0 ∀k 0. (3.16) Comparing Eqs. 3.15 and 3.16, we infer that y 0 = 0. Therefore, The extremality of u * , v * implies that y 0 ∈ C 1 (T ) is a nodal solution of Eq. 1.1.
So, we can now state the first multiplicity theorem for problem (1.1).

Theorem 3.7 If hypotheses H (a)
and H (f ) 2 hold, then problem (1.1) has at least three nontrivial solutions As we already mentioned, hypothesis H (f ) 2 (iii) implies that the reaction f (t, ·) near ζ = 0 exhibits a "concave" term. We can relax this restriction and allow nonlinearities with more general growth near ζ = 0, provided that we restrict the growth of ζ −→ a(ζ ).
Note that this new growth condition on σ implies that The previous analysis concerning nontrivial solutions of constant sign remains valid. What changes is the proof of the existence of a nodal solution.
Also range γ + ⊆ W + . Therefore σ γ + < 0. (3.24) In a similar fashion, we produce a continuous path γ − in W which connects −ε u 0 and v * and such that σ γ − < 0. So, we can now state the second multiplicity theorem for problem (1.1).