Multiple positive solutions for Schrödinger problems with concave and convex nonlinearities

In this paper, we consider the multiplicity of positive solutions for a class of Schrödinger equations involving concave-convex nonlinearities in the whole space. With the help of the Nehari manifold, Ekeland variational principle and the theory of Lagrange multipliers, we prove that the Schrödinger equation has at least two positive solutions, one of which is a positive ground state solution.


Introduction
The method of lower and upper solutions traces as far back as 1886, with Peano's work [22]. Despite that, the existence of extremal solutions and their relation to lower and upper solutions continue to be the subject matter of many research papers on ordinary differential equations; for instance [3,4,11,15]. In recent years, special attention has been devoted to the question of solutions for discontinuous nonlinear differential equations; see e.g. [1,2,10,14]. In this regard, a few steps have been done towards the development of the corresponding theory of extremal solutions for measure differential equations [19].
Measure differential equations, as introduced in [6], are integral equations featuring the Kurzweil-Stieltjes integral. These equations are known to generalize other types of equations, such as classical differential equations, equations with impulses, or dynamic equations on time scales; see [7,20].
Remark 2.2. Given a function f ∈ G([a, b]) and an arbitrary ε > 0, we will denote by D f,ε the division whose existence is guaranteed by Theorem 2.1.
Compactness in the space of regulated functions is connected with the following notion: is said to be equiregulated if for every ε > 0 and every t 0 ∈ [a, b] there exists δ > 0 such that: In the lines of Theorem 2.1, we have the following characterization of equiregulated sets of functions.
Next theorem is the analogue of the Arzelá-Ascoli theorem in the space of regulated functions.
and |f (a)| ≤ M for all f ∈ A. By Lemma 2.4 the set A is equiregulated, and obviously {f (t) : f ∈ A} is bounded for each t ∈ I. Thus, in this case, A is relatively compact.
The following result is concerned with pointwise supremum of regulated functions and, to the best of our knowledge, it is not available in the literature. Proof. From Theorem 2.5, we know that {f (t) : f ∈ A} is bounded for each t ∈ [a, b]; therefore, the function ξ is well defined. Moreover, since A is equiregulated, by Lemma 2.4, given ε > 0 there exists a division D : a = α 0 < α 1 < · · · < α ν(D) = b such that for j ∈ {1, . . . , ν(D)} we have |f (t) − f (s)| < ε for all s, t ∈ (α j−1 , α j ), f ∈ A.
In what follows, given functions f : [a, b] → R n and g : [a, b] → R, the Kurzweil-Stieltjes integral of f with respect to g on [a, b] will be denoted by b a f (s) dg(s), or simply b a f dg. Such an integral has the usual properties of linearity, additivity with respect to adjacent subintervals, as well as the properties to be presented next. The interested reader we refer to [23] or [26].
The following result is known as Hake property.
then b a f dg exists and equals A.
2. If the integral t a f dg exists for every t ∈ [a, b), and A ∈ R n is such that then b a f dg exists and equals A.
Next theorem summarizes some properties of the indefinite Kurzweil-Stieltjes integral.
is regulated and satisfies 3 Vectorial measure differential equations Measure differential equations, in the sense introduced in [6], are integral equations of the form where the integral is understood as the Kurzweil-Stieltjes integral with respect to a nondecreasing function g : I → R and the function f takes values in R n , n ∈ N. Herein we propose a more general version of such an equation where not only f can be a vectorial function but also the integrator g. More precisely, we are interested on equations where I = [t 0 , t 0 + L], y 0 ∈ R n , f : I × R n → R n and g : I → R n . The integral on the right-hand side is simply a notation for a 'component-by-component' integration process in the Kurzweil-Stieltjes sense, that is, writing y 0 = (y 0,1 , . . . , y 0,n ), y = (y 1 , . . . , y n ), f = (f 1 , . . . , f n ), g = (g 1 , . . . , g n ), equation (3.1) corresponds to a systems of n scalar equations, each of which reads as follows This interpretation of the integral in (3.1) is justified by the following vectorial equation where for each t ∈ I, G(t) ∈ R n×n is the diagonal matrix In the case when f (t, y(t)) = y(t), equation (3.3) becomes a particular case of the so-called generalized linear differential equation; a branch of Kurzweil equations theory which has been extensively investigated in [23,26]. Clearly, by taking g i = g : I → R for all i ∈ {1, . . . , n}, from (3.1) we retrieve the notion of measure differential equation introduced in [6]. To avoid any misunderstanding, equations of the type (3.1) will be called vectorial measure differential equations. That said, we define the concept of solution for vectorial measure differential equations. In view of Theorem 2.9, we can see that a solution of (3.1) somehow shares the discontinuity points of g.
Next we introduce the key concepts related to the question of extremal solutions. For that, we consider a partial ordering in R n as follows: given two vectors x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ), we write x ≤ y if x i ≤ y i for each i ∈ {1, . . . , n}. Naturally, for functions α, β : I → R n , we write α ≤ β provided α(t) ≤ β(t) for every t ∈ I. Moreover, if α, β ∈ G(I, R n ) are such that α ≤ β, then we define the functional interval When we want to emphasize that we are considering (lower/upper) solutions which belong to a certain [ α, β], we say that ϕ is a (lower/upper) solution between α and β.
The extremal (greatest and least) solutions to vectorial measure differential equations are defined in the standard way considering the aforementioned ordering, that is, if y : I → R n is a solution of (3.1) we say that: • y is the greatest solution of (3.1) on I if any other solution x : I → R n satisfies x ≤ y; • y is the least solution of (3.1) on I if any other solution x : I → R n satisfies y ≤ x.
In the following sections we will investigate the existence of extremal solutions for (3.1) between lower and upper solutions.
Symmetrically, an upper solution of (3.1) is a function β ∈ G(I, R n ) such that y 0 ≤ β(t 0 ) and Obviously, the reverse inequalities hold for upper solutions.

An existence result for the scalar case
In this section, we turn our attention to the scalar case of equation (3.1), that is, equations of the form where x 0 ∈ R, f : I × R → R and g : I → R is nondecreasing and left-continuous. The existence of the greatest and the least solutions for (4.1) has been already investigated in [19]. In this section, we address one of the questions posed in [19], namely, the existence of (extremal) solutions between given lower and upper solutions. Our result somehow generalizes what is available in the classical theory of ordinary differential equations (cf. [13]) as the function f is not required to be continuous with respect to the first variable.
For the convenience of the reader we will recall the main results in [19]. Given a set B ⊆ R, we consider the following conditions: (C2) There exists a function M : I → R, which is Kurzweil-Stieltjes integrable with respect to for every y ∈ B and [u, v] ⊆ I.
(C3) For each t ∈ I, the mapping y → f (t, y) is continuous in B.
Next lemma is a important tool for dealing with conditions above (see [19,Lemma 3.1]).
If equation (4.1) has a solution on I, then it has the greatest solution x * and the least solution x * on I. Moreover, for each t ∈ I we have Now, we will investigate the existence of extremal solutions for equation (4.1) provided a lower and an upper solutions are known and well-ordered.
Theorem 4.4. Suppose that (4.1) has a lower solution α and an upper solution β such that α(t) ≤ β(t) for all t ∈ I. Assume that the following conditions hold: (H2) There exists a function M : for every y ∈ E and [u, v] ⊆ I.
(H3) For each t ∈ I, the mapping y → f (t, y) is continuous in E.
(H4) For each t ∈ I the mapping is nondecreasing.
Let us define f : and consider the modified problem Clearly, (H3) ensures that f satisfies (C3) with B = R. To show that f satisfies both conditions (C1) and (C2) with B = R, let y ∈ R and put m(t) = max{min{y, β(t)}, α(t)}, t ∈ I. Thus, m ∈ G(I, E) and f (t, y) = f (t, m(t)) for every t ∈ I. Note that Lemma 4.1 and conditions (H1)-(H3) imply that In summary, f satisfies the conditions of Theorem 4.2 and we conclude that (4.5) has a solution defined on the whole of I. The existence of the greatest solution x * and the least solution x * of (4.5) is then a consequence of Theorem 4.3 and assumption (H4). It only remains to show that if x : I → R is an arbitrary solution of equation (4.5), then α(t) ≤ x(t) ≤ β(t), t ∈ I, thus proving that x is a solution of (4.1) and the functions x * and x * are the intended extremal solutions between α and β. Reasoning by contradiction, assume that there exists some t 1 ∈ (t 0 , t 0 + L] such that Using (3.5) and the fact that g is left-continuous, we get ∆ − α(t 2 ) ≤ 0, that is, Hence, we must have α(t 2 ) ≤ x(t 2 ) and consequently t 2 < t 1 . We will show that this leads to a contradiction with (4.6). First, observe that The definition of t 2 implies that x(t) < α(t), t ∈ (t 2 , t 1 ]; thus, by Theorem 2.8 we have Combining this equality with (4.7) we obtain At this point we need to distinguish two cases regarding the value of f . If ). Since condition (H4) implies that The contradiction again follows from condition (H4), now taking into account α(t 2 ) ≤ β(t 2 ). The proof that x ≤ β on I is analogous and we omit it. Moreover, equalities (4.3) and (4.4) follow from Theorem 4.3.

Extremal solutions for vectorial measure differential equations
Our goal is to extend the results from Section 4 to the vectorial equation where y 0 ∈ R n , f : I × R n → R n and g : I → R n . Herein, we assume that g = (g 1 , . . . , g n ) is nondecreasing and left-continuous, that is, for each i ∈ {1, . . . , n}, the function g i : I → R is nondecreasing and left-continuous. Like in the theory of multidimensional equations, the function f is required to be quasimonotone. Recall that a function f is quasimonotone nondecreasing in a set E ⊆ I × R n if given t ∈ I and vectors x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) such that (t, x), (t, y) ∈ E, the following holds: In what follows, e i , i ∈ {1, . . . , n}, denotes the vector in R n whose i-th term is 1 and all others are zero.
Theorem 5.1. Suppose that (5.1) has a lower solution α and an upper solution β such that α ≤ β and assume that f is quasimonotone nondecreasing Furthermore, assume that the following conditions hold: for every η ∈ [ α, β] and [u, v] ⊆ I.
Fix an arbitrary i ∈ {1, . . . , n} and define the function Φ i : which shows that β i : I → R is an upper solution of the scalar problem Using a similar argument, we can show that for any lower solution η of (5.1) such that η ∈ A, the function η i : I → R is a lower solution of (5.7) between α i and β i . Noting that Φ i satisfies the conditions of Theorem 4.4, it follows that (5.7) has the greatest solution x * i : I → R between α i and β i , and by (4.3) η i (t) ≤ x * i (t), t ∈ I, for any η ∈ A lower solution of (5.1). Since the argument is valid for each i ∈ {1, . . . , n}, we construct a function x * = (x * 1 , . . . , x * n ), and obviously ξ * ≤ x * . The quasimonotonicity of f yields for each i ∈ {1, . . . , n}, that is, x * is a lower solution of (5.1) in [ α, β], and for every [u, v] ⊆ I and i ∈ {1, . . . , n}. This shows that x * satisfies (5.4) and (5.5). Thus x * ∈ A, and the definition of ξ * implies x * ≤ ξ * . In summary, ξ * = x * and Therefore, ξ * is a solution of (5.1), and, by (5.6) it is the greatest one in [ α, β].
The proof of the existence of the least solution y * as well the validity of (5.3) is analogous.
6 Extremal solutions for vectorial measure differential equations with functional arguments We will now consider the functional problem where y 0 ∈ R n , f : I × R n × G(I, R n ) → R n and g : I → R n is nondecreasing and left-continuous. We recall that the integral on the right-hand side should be understood as a vectorial Kurzweil-Stieltjes in the sense presented in Section 3.
Equations (6.1) subjected to functional arguments represent a quite general object. It is not hard to see that functional differential equations of the form can be regarded as (6.1) provided the integral of f exists in some sense (in such a case g i corresponds to the identity function for each i). The class of problems covered by (6.1) also includes the so-called measure functional differential equations in the sense introduced in [6]. To see this it is enough to consider f (t, y(t), y) = F (t, y t ) and g = (g, . . . , g), where r > 0, F : I × G([−r, 0], R n ) → R n , g : I → R is nondecreasing and left-continuous, and for each t ∈ I the function y t : [−r, 0] → R n denotes the history or memory of y in [t − r, t], that is, y t (θ) = y(t + θ), θ ∈ [−r, 0].
Unlike the work developed in previous sections, to investigate the extremal solutions for the problem (6.1) we will use a fixed-point approach; namely, the following result which is a consequence of [ Then T has the least fixed point γ * and the greatest fixed point γ * in [ α, β]. Moreover, Proof. We will apply [11, Theorem 1.2.2] assuming X = Y = G(I, R n ) equipped with the supremum norm and the partial ordering defined in Section 3. Given a monotone sequence Next result is the analogue of Theorem 5.1 for functional equations. Note that the notion of lower and upper solutions for equation (6.1) is an obvious extension of Definition 3.2. Indeed, α ∈ G(I, R n ) is a lower solution of (6.1) provided α(t 0 ) ≤ y 0 and while the reverse inequalities are used to define upper solutions of (6.1). Theorem 6.2. Suppose that (6.1) has a lower solution α and an upper solution β such that α ≤ β. For each γ ∈ [ α, β], denote by f γ : I × R n → R n the function defined as f γ (t, x) = f (t, x, γ). Assume that for each γ ∈ [ α, β], the function f γ is quasimonotone nondecreasing in E = {(t, x) ∈ I × R n : α(t) ≤ x ≤ β(t)}. Furthermore, assume that the following conditions hold: (H1) The integral t 0 +L t 0 (f γ ) i (s, η(t)) dg i (s) exists for every γ, η ∈ [ α, β] and i ∈ {1, . . . , n}.
(6.4) Theorem 5.1 guarantees that (6.4) has the greatest solution between α and T η. Since T γ is the greatest solution of (6.4) in [ α, β] it follows that T γ ≤ T η. Hence, T is nondecreasing and Proposition 6.1 yields that T has the greatest fixed point γ * with Naturally, γ * is a solution of (6.1) in [ α, β]. Moreover, it is not hard to see that if γ ∈ [ α, β] is any other solution of (6.1), then γ ≤ T γ. Therefore, by the definition of γ * , we conclude that γ * is the greatest solution of (6.1).
To prove the existence of the least solution for equation (6.1) we proceed in a similar way but redefining the function T so that T γ corresponds to the least solution of equation (6.3).
Assume that α ≤ β and consider the functional interval Further, assume that the following conditions hold:  Note that assumptions (H3) and (H4) do not play a role in Theorem 6.3 as the function f (t, y(t), y) = F (t, y t ) does not depend on y(t).
By setting g(t) = t, equation (6.5) corresponds to the integral form of the retarded functional differential equation Regarding scalar equations (6.6), the existence of solutions between well-ordered lower and upper solutions has been investigated in [24]. Therein, a monotone interactive method is applied in order to obtain the extremal solutions. Although [24] deals with lower/upper solutions which might be discontinuous, the function in the right-hand side, F , is assumed to satisfy the usual Carátheodory conditions. On one hand, in our Theorem 6.3 no continuity is required; however, the monotonicity condition (c) is admittedly stronger than the assumption (P5) stated at [24,Theorem 4].

Applications to Stieltjes differential equations
Stieltjes differential equations are differential systems in which the usual notion of derivative is replaced by a differentiation process with respect to a given monotone function. The basic theory for such equations has been established in [9,17]. In this work, we will consider vectorial Stieltjes differential equations of the form y g (t) = f (t, y(t)) for g-a.a. t ∈ I, y(t 0 ) = y 0 , (7.1) where I = [t 0 , t 0 + L], y 0 ∈ R n , f : I × R n → R n and g : I → R n with g = (g 1 , . . . , g n ) such that, for each i ∈ {1, . . . , n}, g i : I → R is nondecreasing and left-continuous. The problem described by (7.1) should be understood as the following system of Stieltjes differential equations: For a thorough study of the Stieltjes derivative which appears in (7.2) we refer to [17,9]. We remark that the equations studied in [9] are contained in (7.1), corresponding to the particular choice g i = g : I → R for all i ∈ {1, . . . , n}. The Stieltjes equations in [9] were investigated in the space AC g (I) of functions absolutely continuous with respect to g nondecreasing and left-continuous. Recall that a function y ∈ AC g (I) if for every ε > 0 there exists δ > 0 such that m j=1 |y(b j ) − y(a j )| < ε for any family {(a j , b j )} of disjoint subintervals of I satisfying m j=1 (g(b j ) − g(a j )) < δ. Extending the notion of solution found in [9], we will look for solutions of the vectorial problem (7.1) in the space AC g (I) = AC g 1 (I) × · · · × AC gn (I), where g = (g 1 , . . . , g n ) : I → R n is a nondecreasing left-continuous function.
Definition 7.1. A solution of equation (7.1) is a function y ∈ AC g (I) such that (7.2) holds.
As a consequence of the Fundamental Theorem of Calculus for the Lebesgue-Stieltjes integral, [17], we have the following lemma.
If y ∈ AC g (I) is a solution of (7.1), then where the integral stands for the Lebesgue-Stieltjes integral with respect to the Lebesgue-Stieltjes measure µ g i induced by g i . Conversely, if y = (y 1 , . . . , y n ) : I → R n satisfies (7.3), then y ∈ AC g (I) and it solves the vectorial Stieltjes differential equation (7.1).
Using the lemma above and recalling the relation between Lebesgue-Stieltjes and Kurzweil-Stieltjes integrals, [21], one can show that a solution of (7.1) is also a solution of the vectorial measure differential equation (3.1).
In [18], it is shown that, under very general assumptions, the integral equation (4.1) is equivalent to y g (t) = f (t, y(t)) m g -a.e., y(t 0 ) = y 0 , where m g stands for the Thomson's variational measure (see S 0 − µ g in [25]) induced by a function g : I → R. In the case when g is nondecreasing, as proved in [5], the variational measure m g corresponds to the Lebesgue-Stieltjes outer measure µ * g . Therefore, if E ⊂ I and m g (E) = 0, then µ * g (E) = 0 and, consequently, E is Lebesgue-Stieltjes measurable with µ g (E) = 0. Accordingly, a solution of (7.4) also satisfies equation y g (t) = f (t, y(t)) for g-a.a. t ∈ I, y(t 0 ) = y 0 , where y ∈ AC g (I) if and only if f (·, y(·)) is integrable on I with respect to g in the Lebesgue-Stieltjes sense. Therefore, along similar lines of the results in [18], we can draw a correspondence between the solutions of and the solutions of Having all this in mind, based on results of previous sections, we can establish the existence of extremal solutions for Stieltjes differential equations (7.1). Note that extremal solutions to (7.1) are defined in the obvious way in regard to Definition 7.1. We now introduce the concepts of lower and upper solutions for this problem. Definition 7.3. A lower solution of (7.1) is a function α ∈ AC g (I) such that α(t 0 ) ≤ y 0 and (α i ) g i (t) ≤ f i (t, α(t)) for g i -a.a. t ∈ I, i ∈ {1, . . . , n}. (7.5) Analogously, β ∈ AC g (I) is an upper solution of (7.1) if y 0 ≤ β(t 0 ) and Remark 7.4. Every lower solution of (7.1) is also a lower solution of (4.1). Indeed, given a lower solution α of (7.1), since α i ∈ AC g i (I) for each i ∈ {1, . . . , n}, by [9, Theorem 5.1], for every [u, v] ⊆ I we have where the integral stands for the Lebesgue-Stieltjes integral with respect to the Lebesgue-Stieltjes measure µ g i induced by g i . Therefore, (7.5) implies Recall that Lebesgue-Stieltjes integrability implies Kurzweil-Stieltjes integrability, [21]. This, together with the fact that g i is left-continuous, ensures that the Lebesgue-Stieltjes integral on right-hand side coincides with the Kurzweil-Stieltjes integral v u f i (s, α(s)) dg i (s), see [21]. Since functions in the space AC g i (I) have bounded variation ([9, Proposition 5.2]), we conclude that α is a lower solution of the integral equation (4.1).
Similar arguments show that every upper solution of (7.1) is also an upper solution of (4.1).
In [16], extremal solutions for (7.1) have been studied in the scalar case. In order to apply the results of previous sections to investigate the solutions of the vectorial problem (7.1) we will need the following lemma which corresponds to a particular case of [18,Lemma 2.22]. Theorem 7.6. Suppose that (7.1) has a lower solution α and an upper solution β such that α ≤ β. Assume that f is quasimonotone nondecreasing in E = {(t, x) : α(t) ≤ x ≤ β(t)} and that the following conditions hold: (ii) for each η ∈ [ α, β] and for g i -a.a. t ∈ I, the mapping is continuous; (iii) for every r > 0, there exists a function h i,r : I → R + , which is Lebesgue-Stieltjes integrable with respect to g i , such that |f i (t, x)| ≤ h i,r (t), for g i -a.a. t ∈ I, for every x ∈ R n , x ≤ r.
Then equation ( Proof. For each i ∈ {1, . . . , n}, let N i ⊂ I be a g i -null set such that both conditions (ii) and (iii) Let U = (U 1 , . . . , U n ) : I × R n → R n and consider the modified problem It follows from Lemma 7.5 that a solution of (7.6) is also a solution of (3.1). Moreover, (A) guarantees that the integrals in ( To prove (H2), take r = max{ α ∞ , β ∞ } and let h i,r : I → R + , i ∈ {1, . . . , n}, be the corresponding function in (A)(iii). Since and both functions are integrable with respect to g i in the sense of Kurzweil-Stieltjes, we get for every [u, v] ⊆ I and x ∈ R n with x ≤ r. Proceeding as in the proof of Lemma 3.1 in [19], we can show that the inequality above still holds if we consider regulated functions x : I → R n with x ∞ ≤ r. Thus, (H2) follows and this concludes the proof.
Consider now the functional problem, y g (t) = f (t, y(t), y) for g-a.a. t ∈ I, y(t 0 ) = y 0 , where y 0 ∈ R n , f : I × R n × G(I, R n ) → R n and g : I → R n is nondecreasing and left-continuous. Naturally, as before, (7.7) denotes a system of Stieltjes differential equations subject to functional arguments; namely: A solution of this functional equation is defined analogously to those of problem (7.1), and so are the upper and lower solutions. Extremal solutions for (7.7) were recently investigated in [16] in the scalar case. Next, applying Theorem 6.2, we present a result for (7.7) in its general formulation. Such a result relies on a correspondence between (7.7) and (6.1) which can be obtained from an extension of the argument used in [18].

A simple model for a bacteria population with variable carrying capacity
Consider an open tank which contains an initial amount of water reaching a level of L meters high and assume that the changes on the level of water are exclusively caused by evaporation as a result of the effect of the sun. According to this, during the day the level of water will change, whereas it will remain constant during night hours. Now consider a bacteria population whose resources depend directly on the volume of water. This means that the carrying capacity (that is, the number of bacteria that can be supported indefinitely in the tank) will be dependent on the level of water: the higher the level of water is, the bigger the carrying capacity will be. Finally, we will also assume that every morning, the tank is refilled until a certain level depending on the population of bacterias at that time.
We want to design a mathematical model for w(t), the water height at time t > 0, and p(t), the bacteria population at time t > 0, under the previous assumptions. For the latter, we will consider a logistic model where the carrying capacity will be given by a nondecreasing function N : R → R depending on w(t). Hence, the population p at time t is represented by the equation with r > 0 being the reproduction rate of the population.
Clearly, the map f defined in (8.2) is quasimonotone nondecreasing in I×R 2 , and in particular in E = {(t, x) : α(t) ≤ x ≤ β(t)}. Moreover, it is easy to check that f satisfies hypotheses (A)-(B) of Theorem 7.6, therefore the problem (8.1) has the extremal solutions between α and β.
So far, we have only used the fact that N (w) is a nondecreasing function of w, so no matter if it is continuous or not, our theory applies. However, in some cases we may find it reasonable to allow the carrying capacity N (w) to be piecewise constant because very small changes in the water level could have no influence on the carrying capacity. A simple example appears when we consider N to be the floor function, N (t) = t . As we mentioned before, W (t) is a solution of w g (t) = F (t, p(t), w(t)), w(0) = L.