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

https://doi.org/10.15388/NA.2019.3.8 Nonlinear Analysis: Modelling and Control, Vol. 24, No. 3, 447–461 eISSN: 2335-8963 ISSN: 1392-5113 Positive solutions for φ-Laplace equations with discontinuous state-dependent forcing terms Radu Precupa, Jorge Rodríguez-Lópezb,1 aDepartment of Mathematics, Babe¸s-Bolyai University, 400084 Cluj-Napoca, Romania r[email protected] bInstituto de Matemáticas, Universidade de Santiago de Compostela, 15782, Facultade de Matemáticas, Campus Vida, Santiago, Spain [email protected] Received: October 24, 2018 / Revised: November 21, 2018 / Published online: April 19, 2019 Abstract. This paper concerns the existence, localization and multiplicity of positive solutions for aφ-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. Keywords: discontinuous differential equation, φ-Laplacian problem, positive solution, fixed point, multivalued map, infinitely many solutions. 1 Introduction In this paper, we establish new existence, localization and multiplicity results of positive solutions for the problem −φu0)0=f(t, u)a.e. in I:= [0,1], u(0) −αu0(0) = u0(1) = 0,(1) 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). 1The author was financially supported by Xunta de Galicia Scholarship ED481A-2017/178. c Vilnius University, 2019 448 R. Precup, J, Rodríguez-López 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 fwas previously studied in [23,24], using Krasnosel’ski˘ ı’s compression-expansion fixed point theorem in cones and a Harnack-type inequality. In the present paper, since the function fmay be discontinuous, we first consider the regularized problem in the Filippov sense [20], namely the boundary value problem for a differential inclusion −φ(u0))0∈F(t, u)a.e. in I, u(0) −αu0(0) = u0(1) = 0,(2) where the multivalued map F:I×R+→ P(R+)is defined as F(t, x) = \ ε>0 coft, Bε(x)∩R+(3) with co standing for the closed convex hull and Bε(x) := [x−ε, x +ε]. Unfortunately, the standard generalization of Krasnosel’ski˘ ı’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 Positive solutions for φ-Laplace equations 449 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 fis 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. 2 The fixed point setting In [17,28], an existence theory for the operator inclusion x∈ΨΦx, (4) where Ψand Φare two single or multivalued operators, was developed. Let Xbe a normed linear space. Let us introduce the following notations: Pfc(X) = A⊂X:Ais nonempty, closed and convex, Pkwc(X) = A⊂X:is nonempty, weakly compact and convex. Also, recall that a multivalued operator Φfrom a subset Dof a normed linear space to an other normed linear space is said to be •upper semicontinuous (usc, for short) on Dif for every closed subset Cof D, the set Φ−(C) = x∈D:C∩Φx 6=∅ is closed in D. •sequentially weakly upper semicontinuous (w-usc, for short) on Dif for every weakly closed subset Cof D, the set Φ−(C)is sequentially closed for the weak topology on D. Now we state the compression-expansion fixed point theorem for inclusion (4). Theorem 1. (See [17, Thm. 2.3].) Let (X, k·k)and Ybe normed linear spaces, and Ka wedge of X. Let Φ:K→Pkwc(Y),Ψ:C→Pfc(K)be two bounded multivalued maps, where C= co({0}∪Φ(K)). Assume that (i) if A⊂K,A= co({0}∪Ψ(co({0} ∪ Φ(A)))), then Ais weakly compact, and Φ,Ψare w-usc on Aand co({0}∪Φ(A)), respectively. In addition, assume that there exist r1, r2>0,r16=r2, and h∈K\{0}such that (ii) x /∈λΨΦx for λ∈(0,1) and x∈Kwith kxk=r1;and x /∈ΨΦx +µh for µ > 0and x∈Kwith kxk=r2. Nonlinear Anal. Model. Control, 24(3):447–461 450 R. Precup, J, Rodríguez-López Then there exists at least one x∈Kwith x∈ΨΦx such that min{r1, r2}⩽kxk⩽max{r1, r2}. 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 (iii) if A⊂K,A= co({0}∪Ψ(co({0}∪Φ(A)))), then Ais compact, and Φ,Ψare usc on Aand co({0}∪Φ(A)), respectively. 3 Main result In this section, we study the existence of positive solutions to problem (1), that is, a function u∈C1(I),u⩾0,u6≡ 0, with u(0) −αu0(0) = u0(1) = 0 such that φ◦u0∈W1,1(I)and −φu0(t)0=ft, u(t)for a.a. t∈I. Equivalently, we will look for fixed points of the integral operator T:P→Pgiven by Tu(t) = αφ−1 1 Z 0 fs, u(s)ds!+ t Z 0 φ−1 1 Z r fs, u(s)ds!dr, (5) where Pis the cone of nonnegative functions in the Banach space of the continuous functions with the maximum norm (C(I),k·k∞). As mentioned above, since fis not necessarily continuous, the operator Tmay 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 Tu(t) = αφ−1 1 Z 0 Fs, u(s)ds!+ t Z 0 φ−1 1 Z r Fs, u(s)ds!dr, (6) where Fstands for the map obtained after “convexification” of the function fas in (3). Notice that the operator Tcan be decomposed as T=ΨΦ, where for every v∈P, Ψv(t) = αφ−1v(0)+ t Z 0 φ−1v(s)ds, t ∈I, and Φv(t) = ΛNFv(t) https://www.mii.vu.lt/NA Positive solutions for φ-Laplace equations 451 with Λw(t) = 1 Zt w(s) ds and the Nemytskii operator NF(u) = v∈L1(I): v(t)∈Ft, u(t)for a.a. t∈I.(7) 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: (H1) The composed function f(·, u(·)) is measurable for every u∈P; (H2) f(t, u)< b on I×R+, and if b=∞, there exist c1, c2∈R+and p⩾1such that f(t, u)⩽c1up+c2for a.a. t∈Iand all u∈R+. Then the Nemytskii operator NF:P→ P(L1(I)) defined as in (7) is an usc map on P from the topology of C(I)to that of L1(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 < ∞. Lemma 2. For each c∈(0,1) and any u∈C1(I),u⩾0, with u(0) −αu0(0) = u0(1) = 0 and φ◦u0nonincreasing in I, one has u(t)⩾Mkuk∞for all t∈[c, 1], where M= (α+c)/(α+ 1). From now on, the point c∈(0,1) is fixed. The essential properties of the operators Φ and Ψfrom above are given by the following theorem involving two subcones of P, namely K1=u∈P:u(t)⩾Mkuk∞for all t∈[c, 1], K2=u∈P:uis nonincreasing, u(1) = 0, and, exclusively, the topology of C(I). Theorem 2. If the function fsatisfies conditions (H1) and (H2), then the operators Φ:K1→Pfc(K2)and Ψ:K2→K1 are well defined;Φis usc and maps bounded sets into relatively compact sets;and Ψis a single-valued continuous operator, which maps bounded sets into relatively compact sets. Proof. Since Φ=ΛNF, it follows from the definition of the operator Λthat Φ(K1)⊂K2. To show that Ψ(K2)⊂K1, take any v∈K2and let u:= Ψ(v). Clearly, u∈P. Also, φ◦u0=v, and so φ◦u0is nonincreasing in I. Moreover, u(0) −αu0(0) = u0(1) = 0. Consequently, by Lemma 2, u(t)⩾Mkuk∞for all t∈[c, 1]. Hence u∈K1, as desired. Nonlinear Anal. Model. Control, 24(3):447–461 452 R. Precup, J, Rodríguez-López In addition, Λ, as a linear operator from L1(I)to C(I), is compact, while in view of Lemma 1, NFis usc from the topology of C(I)to that of L1(I). Thus Φis usc and maps bounded sets into relatively compact sets. Clearly, Φhas convex values. To show that its values are also closed in C(I), take any element u∈K1and any sequence vn∈Φu with vn→vin C(I). Then vn=Λwn for some wn∈ NF(u). From the definition of Fwe have that NF(u)(t)is bounded uniformly with respect to t∈I. As a result, the sequence wnis bounded in Lp(I)for any (fixed) p∈(1,∞). The space Lp(I)(for 1< p < ∞)being reflexive, we may assume without less of generality that wnis weakly convergent in Lp(I)to some w. It is easy to see that w∈NF(u). Then there is a sequence wnof convex combinations of wn, which strongly converges in Lp(I), and consequently in L1(I), to w. From vn=Λwn we deduce that the corresponding sequence vnof convex combinations of vnconverges in C(I)to Λw. But since vn→v, the limit of vnis v. Then v=Λw, where w∈NF(u), which proves that v∈Φu, as wished. Finally, the continuity and the compactness of the operator Ψare 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 fsatisfies conditions (H1), (H2) and (H3) There is a countable number of functions γn∈C1(I)(n∈N)with φ◦γ0 n∈ W1,1(I)and a countable number of closed subintervals Inof Isuch that −φγ0 n(t)0∩Ft, γn(t)⊂ft, γn(t) for a.a. t∈In, n ∈N,(8) and f(t, ·)is continuous on R+\[ {n:t∈In}γn(t)for a.a. t∈I. (9) In addition, assume that there exist 0< r1, r2,r16=r2, and ε > 0such that αφ−1 1 Z 0 Γε r1(s) ds!+ 1 Z 0 φ−1 1 Z r Γε r1(s) ds!dr⩽r1,(10) αφ−1 1 Z c Γr2,ε(s) ds!+ 1 Z c φ−1 1 Z r Γr2,ε(s) ds!dr⩾r2,(11) where Γε r1(s) = max x∈[0, r1+ε]f(s, x)and Γr2,ε(s) = min x∈[(r2−ε)M, r2+ε]f(s, x). Then problem (1) has at least one positive solution usuch that min{r1, r2}⩽kuk∞⩽max{r1, r2}.(12) https://www.mii.vu.lt/NA Positive solutions for φ-Laplace equations 453 Proof. We apply Theorem 1. In virtue of Theorem 2, it only remains to prove that the operator T=ΨΦ satisfies the compression-expansion conditions as in (ii). We first show that kvk∞⩽r1for all v∈Tuand u∈K1with kuk∞=r1, which implies that u /∈λTufor all λ∈(0,1) and u∈K1with kuk∞=r1. Assume the contrary. Then there exists v∈Tuand u∈K1with kuk∞=r1such that r1<kvk∞. Notice that for any ε > 0, if w∈ NF(u)and kuk∞=r1, then w(s)⩽max x∈[0, r1+ε]f(s, x) =: Γε r1(s)for all s∈I. Hence, by the fact that v∈Tuand (10), kvk∞⩽αφ−1 1 Z 0 Γε r1(s) ds!+ 1 Z 0 φ−1 1 Z r Γε r1(s) ds!dr⩽r1, which yields the contradiction r1< r1. Next, we have to show that r2⩽kvk∞for all v∈Tuand u∈K1with kuk∞=r2, which implies that u /∈Tu+µfor all µ > 0and u∈K1with kuk∞=r2. The proof is similar and is based on the fact that for every ε > 0, if w∈ NF(u)and kuk∞=r2, w(s)⩾Γr2,ε(s)for all s∈[c, 1]. The details are left to the reader. Therefore, Theorem 1 applies and yields the existence of a fixed point u∈Pfor the operator Tsatisfying (12). Then −φu0(t)0∈Ft, u(t)for a.a. t∈I. (13) Now we prove that u(in fact, any fixed point of T) solves the initial discontinuous problem (1). To this aim, define Jn:= t∈In:u(t) = γn(t), n ∈N. Clearly −φu0(t)0=−φγ0 n(t)0for a.a. t∈Jn. Nonlinear Anal. Model. Control, 24(3):447–461 454 R. Precup, J, Rodríguez-López Hence, by (13), −φγ0 n(t)0∈Ft, u(t)=Ft, γn(t)for a.a. t∈Jn. This, based on condition (8), implies −φγ0 n(t)0=ft, γn(t)for a.a. t∈Jn, equivalently, −φu0(t)0=ft, u(t)for a.a. t∈Jn. Thus usatisfies the initial discontinuous differential equation a.e. in J=Sn∈NJn. Finally, from (9) one has Ft, u(t)=ft, u(t) for t∈I\J. This together with (13) shows that ualso satisfies the initial discontinuous differential equations a.e. in I\J. Therefore, usolves (1) in I. Remark 2. If the function fis nondecreasing with respect to the second variable, then conditions (10) and (11) can be written as αφ−1 1 Z 0 f(s, r1+ε) ds!+ 1 Z 0 φ−1 1 Z r f(s, r1+ε) ds!dr⩽r1, αφ−1 1 Z c fs, M(r2−ε)ds!+ 1 Z c φ−1 1 Z r fs, M(r2−ε)ds!dr⩾r2, 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 fis nondecreasing with respect to the second variable, the existence of two numbers r1and r2 satisfying (10) and (11) is guaranteed by any one of the following two conditions: (a) lim inf x→0 αφ−1(R1 0f(s, x) ds) + R1 0φ−1(R1 rf(s, x) ds) dr x<1, lim sup x→∞ αφ−1(R1 cf(s, Mx) ds) + R1 cφ−1(R1 rf(s, Mx) ds) dr x>1; (14) (b) lim inf x→∞ αφ−1(R1 0f(s, x) ds) + R1 0φ−1(R1 rf(s, x) ds) dr x<1, lim sup x→0 αφ−1(R1 cf(s, Mx) ds) + R1 cφ−1(R1 rf(s, Mx) ds) dr x>1. (15) https://www.mii.vu.lt/NA Positive solutions for φ-Laplace equations 455 Observe that the first case is only possible if a= +∞and b= +∞. Otherwise, if a < +∞, then φ−1is bounded, and if b < +∞, then f < b, so the numerator being bounded, lim sup x→∞ αφ−1(R1 cf(s, Mx) ds) + R1 cφ−1(R1 rf(s, Mx) ds) dr x= 0. 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 fto 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: (i) −(φ(γ0 n(t)))0=f(t, γn(t)) for a.a. t∈In; (ii) {−(φ(γ0 n(t)))0}/∈F(t, γn(t)) for a.a. t∈In. In particular, alternative (ii) is satisfied if there exist δ, ε > 0such that −φγ0 n(t)0+δ⩽f(t, y)for a.a. t∈Inand all y∈γn(t)−ε, γn(t) + ε,(16) or −φγ0 n(t)0−δ⩾f(t, y)for a.a. t∈Inand all y∈γn(t)−ε, γn(t) + ε.(17) Observe also that conditions (16) and (17) recall the notion of lower and upper solutions for the differential equation −(φ(u0))0=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 −u0 √1 + u020 =f(t, u) := 3 √ue−u+1 2cos2 1 u+t a.e. in I, u(0) = u0(1) = 0, (18) where [x]denotes the integer part of x. Here, φ:R→(−1,1) is given by φ(τ) = τ √1 + τ2and φ−1(τ) = τ √1−τ2. Also, notice that f(t, u)<1. For this example, γn=−t+1 nand In=0,1 n, n ∈N. Nonlinear Anal. Model. Control, 24(3):447–461