Positive solutions for φ-Laplace equations with discontinuous state-dependent forcing terms

Abstract. This paper concerns the existence, localization and multiplicity of positive solutions for a φ-Laplacian problem with a perturbed term that may have discontinuities in the state variable. First, the initial discontinuous differential equation is replaced by a differential inclusion with an upper semicontinuous term. Next, the existence and localization of a positive solution of the inclusion is obtained via a compression-expansion fixed point theorem for a composition of two multivalued maps, and finally, a suitable control of discontinuities allows to prove that any solution of the inclusion is a solution in the sense of Carathéodory of the initial discontinuous equation. No monotonicity assumptions on the nonlinearity are required.


Introduction
In this paper, we establish new existence, localization and multiplicity results of positive solutions for the problem where α 0, φ : (−a, a) → (−b, b) is an increasing homeomorphism such that φ(0) = 0, 0 < a, b ∞, and the function f : I × R + → R + may have discontinuities even with respect to the second variable.Also, we achieve a multiplicity result concerning the existence of infinitely many solutions to problem (1).
Differential equations with discontinuous nonlinearities (discontinuous differential equations, for short) arise from mathematical modeling of real processes from physics, engineering, biology, medicine, economics etc., whose dynamics are designed by discontinuous feedback controllers.For example, the regulation of temperature in a room is achieved discontinuously by switching on and off the cooling system as operated by a thermostat.In mechanics, such a process is the movement in presence of dry friction [26], while in population dynamics, it is the harvesting whose intensity is changed depending on some population thresholds.A nonstandard example of discontinuous differential equation can be found in [14] modeling opinion dynamics.In mathematics, discontinuous differential equations appear in control theory, where, even when a system is controllable, it may fail to admit a continuous feedback, which stabilizes it [1,16,20].
Discontinuous differential equations have been studied by many authors, see, e.g., [4,5,8,10,15,25,32] and the references therein.To this aim, several techniques have been used, such as methods of fixed point theory [19,27], lower and upper solution techniques [32], or methods of nonsmooth critical point theory [5,7,8].In connection with the investigation methods, several notions of solution have been defined, most of them as solutions of some differential inclusions, see [22,31].
Also, in the last decades, φ-Laplacian equations have been extensively studied by different authors and a variety of tools, see, e.g., [3,12,30].However, not many results on discontinuous φ-Laplacian equations are known in the literature.The existing ones are based on monotonicity hypotheses for nonlinearities [11] or use solutions in the sense of set-valued analysis, mainly, as Filippov or Krasovskij solutions; see [2,9].Compared to these results, in our case, the solutions are in the Carathéodory sense, and no monotonicity assumptions are required.
Problem (1) with a continuous nonlinearity f was previously studied in [23,24], using Krasnosel'skiȋ's compression-expansion fixed point theorem in cones and a Harnack-type inequality.
In the present paper, since the function f may be discontinuous, we first consider the regularized problem in the Filippov sense [20], namely the boundary value problem for a differential inclusion where the multivalued map F : I × R + → P(R + ) is defined as with co standing for the closed convex hull and Unfortunately, the standard generalization of Krasnosel'skiȋ's fixed point theorem to upper semicontinuous multivalued maps with convex values, due to Fitzpatrick and Petryshyn [21], is not applicable to the integral operator associated to problem (2).The reason is that the values of the integral operator are not convex in general due to the https://www.mii.vu.lt/NA nonlinearity of φ.To overcome this difficulty, we will apply a compression-expansion fixed point theorem established in [17] for the composition of two multivalued operators (see also the coincidence point theorems in [6]).
After obtaining and localizing a solution of the regularized problem, we shall concentrate on proving that any solution of the regularized problem is also a solution of the initial discontinuous equation in the sense of Carathéodory.We shall succeed in this, by using the technique from [15,18,19,27,29], under the assumption that the function f is discontinuous over the graphs of a countable number of curves satisfying some "transversality" condition.
Finally, for problems with nonlinearities having excessive oscillations towards zero or infinity, by using the localization result, we are able to emphasize the existence of infinitely many positive solutions.

The fixed point setting
In [17,28], an existence theory for the operator inclusion where Ψ and Φ are two single or multivalued operators, was developed.Let X be a normed linear space.Let us introduce the following notations: A is nonempty, closed and convex , P k w c (X) = A ⊂ X: is nonempty, weakly compact and convex .
Also, recall that a multivalued operator Φ from a subset D of a normed linear space to an other normed linear space is said to be ), then A is weakly compact, and Φ, Ψ are w-usc on A and co({0} ∪ Φ(A)), respectively.
Then there exists at least one x ∈ K with x ∈ Ψ Φx such that Remark 1.Since any usc map on a compact set is sequentially w-usc, Theorem 1 remains true if instead of (i) we assume condition ), then A is compact, and Φ, Ψ are usc on A and co({0} ∪ Φ(A)), respectively.

Main result
In this section, we study the existence of positive solutions to problem (1), that is, a func- Equivalently, we will look for fixed points of the integral operator T : P → P given by where P is the cone of nonnegative functions in the Banach space of the continuous functions with the maximum norm (C(I), • ∞ ).
As mentioned above, since f is not necessarily continuous, the operator T may be discontinuous, and the usual compression-expansion-type results are not applicable.This is the motivation to consider inclusion (2) and to look for solutions of this problem by means of the multivalued operator T : P → P(P ) defined as where F stands for the map obtained after "convexification" of the function f as in (3).
Notice that the operator T can be decomposed as where for every v ∈ P , First, let us mention the following result about the upper semicontinuity of the Nemytskii operator (for details, see [13,29]).
Lemma 1. Assume that the function f : I × R + → R + satisfies the following conditions: Then the Nemytskii operator N F : P → P(L 1 (I)) defined as in ( 7) is an usc map on P from the topology of C(I) to that of L 1 (I).
In order to apply Theorem 1, we need the following Harnack-type inequality established in [23,24] for the case a = ∞.Notice that with the same proof, the result still holds true for a < ∞.
From now on, the point c ∈ (0, 1) is fixed.The essential properties of the operators Φ and Ψ from above are given by the following theorem involving two subcones of P , namely and, exclusively, the topology of C(I).
Theorem 2. If the function f satisfies conditions (H1) and (H2), then the operators are well defined; Φ is usc and maps bounded sets into relatively compact sets; and Ψ is a single-valued continuous operator, which maps bounded sets into relatively compact sets.
Proof.Since Φ = ΛN F , it follows from the definition of the operator In addition, Λ, as a linear operator from L 1 (I) to C(I), is compact, while in view of Lemma 1, N F is usc from the topology of C(I) to that of L 1 (I).Thus Φ is usc and maps bounded sets into relatively compact sets.
Clearly, Φ has convex values.To show that its values are also closed in C(I), take any element u ∈ K 1 and any sequence v n ∈ Φu with v n → v in C(I).Then v n = Λw n for some w n ∈ N F (u). From the definition of F we have that N F (u)(t) is bounded uniformly with respect to t ∈ I.As a result, the sequence w n is bounded in L p (I) for any (fixed) p ∈ (1, ∞).The space L p (I) (for 1 < p < ∞) being reflexive, we may assume without less of generality that w n is weakly convergent in L p (I) to some w.It is easy to see that w ∈ N F (u). Then there is a sequence w n of convex combinations of w n , which strongly converges in L p (I), and consequently in L 1 (I), to w.From v n = Λw n we deduce that the corresponding sequence v n of convex combinations of v n converges in C(I) to Λw.But since v n → v, the limit of v n is v. Then v = Λw, where w ∈ N F (u), which proves that v ∈ Φu, as wished.
Finally, the continuity and the compactness of the operator Ψ are standard consequences of Lebesgue's dominated convergence and Ascoli-Arzela's theorems.Now we are ready to state and prove the main result about the existence and localization of positive solutions to the discontinuous problem (1).
Theorem 3. Assume that the function f satisfies conditions (H1), (H2) and (H3) There is a countable number of functions In addition, assume that there exist 0 < r 1 , r 2 , r 1 = r 2 , and ε > 0 such that where Then problem (1) has at least one positive solution u such that https://www.mii.vu.lt/NAProof.We apply Theorem 1.In virtue of Theorem 2, it only remains to prove that the operator T = Ψ Φ satisfies the compression-expansion conditions as in (ii).We first show that v ∞ r 1 for all v ∈ Tu and u ∈ K 1 with u ∞ = r 1 , which implies that u / ∈ λTu for all λ ∈ (0, 1) and u ∈ K 1 with u ∞ = r 1 .
Assume the contrary.Then there exists v ∈ Tu and u ∈ K 1 with u ∞ = r 1 such that r 1 < v ∞ .Notice that for any ε > 0, if w ∈ N F (u) and u ∞ = r 1 , then Hence, by the fact that v ∈ Tu and (10), which yields the r 1 < r 1 .
Next, we have to show that which implies that u / ∈ Tu + µ for all µ > 0 and u ∈ K 1 with u ∞ = r 2 .
The proof is similar and is based on the fact that for every ε > 0, if w ∈ N F (u) and The details are left to the reader.Therefore, Theorem 1 applies and yields the existence of a fixed point u ∈ P for the operator T satisfying (12).Then Now we prove that u (in fact, any fixed point of T) solves the initial discontinuous problem (1).To this aim, define Hence, by (13), This, based on condition (8), implies Thus u satisfies the initial discontinuous differential equation a.e. in J = n∈N J n .Finally, from ( 9) one has This together with (13) shows that u also satisfies the initial discontinuous differential equations a.e. in I \ J. Therefore, u solves (1) in I.
Remark 2. If the function f is nondecreasing with respect to the second variable, then conditions ( 10) and ( 11) can be written as and they are analogous to those considered in [23] for the case of a continuous nonlinearity.

Remark 3 [Asymptotic conditions].
In virtue of Remark 2, if the function f is nondecreasing with respect to the second variable, the existence of two numbers r 1 and r 2 satisfying ( 10) and ( 11) is guaranteed by any one of the following two conditions: (a) https://www.mii.vu.lt/NAObserve that the first case is only possible if a = +∞ and b = +∞.Otherwise, if a < +∞, then φ −1 is bounded, and if b < +∞, then f < b, so the numerator being bounded, The "transversality" condition ( 8) was previously presented in [29] and recalls the notion of viable and inviable curves introduced in [19,27].It allows the function f to be discontinuous over time-dependent sets that generalize the discontinuity sets of [5,7].The meaning of ( 8) is clarified by the next remark, where some sufficient conditions are given.
Remark 4. Assumption ( 8) is satisfied if one of the following two conditions holds: In particular, alternative (ii) is satisfied if there exist δ, ε > 0 such that Observe also that conditions ( 16) and ( 17) recall the notion of lower and upper solutions for the differential equation −(φ(u )) = f (t, u).
To finish this section, we illustrate the applicability of our main result by an example.
Example 1.Consider the differential problem involving the curvature operator in Euclidean space where [x] denotes the integer part of x.Here, φ : R → (−1, 1) is given by Also, notice that f (t, u) < 1.
For this example, Clearly, the function u → f (t, u) is continuous on R + \ {n: t∈In} {γ n (t)} for a.a.t ∈ I.Note that − φ γ n (t) = 0 for a.a.t ∈ I n and n ∈ N, so condition ( 16) is obviously satisfied for δ, ε > 0 small enough since cos 2 n > 0 for all n ∈ N .
Therefore, Theorem 3 ensures the existence of a positive solution of problem ( 18) such that 1/50 u ∞ 1.

Infinitely many solutions
The existence and localization result, Theorem 3, yields multiplicity results for problem (1) when several couples of numbers r 1 and r 2 satisfying conditions ( 10) and (11) exist such that the corresponding intervals (r, R) are disjoint.Taking this into account and using asymptotic conditions, it is possible to derive a multiplicity result concerning the existence of infinitely many positive solutions for problem (1).
holds, then problem (1) has a sequence of positive solutions Proof.Notice that, in view of Remarks 2 and 3, condition (c) ensures that there exist two sequences (r 1,n ) n∈N and (r 2,n ) n∈N such that https://www.mii.vu.lt/NA and for each n ∈ N, the numbers r 1,n and r 2,n satisfy (10) and (11).Therefore, for each n ∈ N, Theorem 3 ensures the existence of a solution u n such that Now (19) and (20) show that the solutions u n are distinct being located in disjoint annular sets, and also that u n ∞ → ∞ as n → ∞.
In a similar way, condition (d) and Theorem 3 ensure the existence of a sequence of positive solutions (u n ) n∈N in the conditions of the statement.
Then, it is easy to verify that f takes nonnegative values and is nondecreasing on R + .Obviously, f is discontinuous at the natural numbers (the points where the integer part function is discontinuous) and so, γ n (t) ≡ n and I n = [0, 1] for n ∈ N.
In the next example, φ is not the identity function, while the right-hand side is a slight modification of that from Example 2.