# A Variational Method for Multivalued Boundary Value Problems

Authors: Droh Arsène Béhi; Assohoun Adjé
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

In this paper, we investigate the existence of solution for differential systems involving aϕ−Laplacian operator which incorporates as a special case the well-known p−Laplacian operator. In this purpose, we use a variational method which relies on Szulkin’s critical point theory. We obtain the existence of solution when the corresponding Euler–Lagrange functional is coercive.


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## Body

## 1. Introduction

This paper is devoted to the study of the following second-order differential systems:(1)−ϕu′t′+εϕut=∇Ft,uta.e.onΩ=0,Tϕu′0,−ϕu′T∈∂ju0,uT,where ε≥0 is fixed, ϕ:ℝN⟶ℝN is a monotone homeomorphism, j:ℝN×ℝN⟶−∞,+∞ is a proper, convex, and lower semicontinuous function, F:Ω×ℝN⟶ℝN is a Caratheodory mapping, continuously differentiable with respect to the second variable and satisfies some usual growth conditions, ∇Ft,x is the gradient of Ft,. at x∈ℝN and ∂j denotes the subdifferential of j in the sense of convex analysis.Second-order differential problems with multivalued boundary value conditions have been studied by many authors. In this direction, we can cite the works of Bader and Papageorgioua [1], Béhi et al. [2], Gasinski and Papageorgiou [4], Jebelean [5], Jebelean and Morosanu [6], and references therein. In [1, 4–6], the authors investigate differential systems driven by a homogeneous p−Laplacian operator while in problem (1), we deal with a nonhomogeneous ϕ−Laplacian operator which incorporates as a special case the p−Laplacian operator. As a consequence, problem (1) is a generalization of the following problem studied, in 2005, by Jebelean and Morosanu [6]:(2)−ϕpu′t′+εϕput=∇Ft,uta.e.onΩ=0,Tϕpu′0,−ϕpu′T∈∂ju0,uT,where ϕp denotes the homogeneous operator p−Laplacian. Indeed, in our work, the ϕ−Laplacian operators are nonhomogeneous and are of the form ϕx=axϕpx with ϕp the p−Laplacian operator, p>1, and a:ℝN⟶ℝ+∗ a continuous map.In order to obtain existence result for problem (1), we will use a variational method which relies on Szulkin’s critical point theory [7].This paper is organized as follows: After introducing notations and preliminary results in Section2, in Section 3 we give a variational approach to problem (1). In Section 4, using some results from Section 3, we prove our main result. In Section 4, we give an example to illustrate the applicability of our result. Finally, Section 5 is reserved for the conclusion.


## 2. Preliminaries

Let us recall some notions and results in the framework of Szulkin’s critical point theory [7] which are needed and also some notations which will be used in the sequel. At this end, W1,p0,T,ℝN is the Sobolev Banach space which will be endowed with the norm:(3)um=mupp+u′pp1/p,where m>0 and .p is the norm on Lp0,T,ℝN defined by(4)up=∫0Tupdt1/p.We denote.∞, the norm on the set C0,T,ℝN:(5)u∞=maxut:t∈0,T.LetX, be a real Banach space and B:X⟶−∞,+∞ be a functional of type(6)B=E+G,where E∈C1X,ℝ and G is proper, convex, and lower semicontinuous (lsc). A point u∈X is said to be a critical point of B if it satisfies the inequality(7)E′u,v−u+Gv−Gu≥0,∀v∈X.A numberc∈ℝ such that B−1c contains a critical point is called a critical value of B.To establish existence results for (1), we will need the following proposition due to Szulkin (see [7] Proposition 1.1).Proposition 1.
IfB satisfies (7), each local minimum point of B is necessarily a critical point of B.The functionalB is said to satisfy the Palais-smale (in short, (PS)) condition if every sequence un⊂X for which Bun⟶c∈ℝ and(8)E′un,v−un+Gv−Gun≥εnv−un,∀v∈X,where εn⟶0 possesses a convergent subsequence.


## 3. A Variational Approach for Problem (1)

Before beginning the variational approach, we make the following hypotheses on the data of problem (1):Ha: a:ℝN⟶ℝ+∗ is a continuous map such that there exists η,M>0 such that(9)η≤ax≤M,∀x∈ℝN.Hϕ: ϕ:ℝN⟶ℝN is a monotone homeomorphism such thata ϕx=axϕpx, ∀x∈ℝN, with ϕp the well-know p−Laplacian operator and ∇Φ=ϕ with Φ:ℝN⟶ℝ of class C1 on ℝN, strictly convex (i.e., Φ is a potential function corresponding to ϕ) and such that Φ0=0;b there exist c1,c2,c3>0 such that(10)c1xp≤Φx≤c2+c3xp,∀x∈ℝN,where. denotes the Euclidean norm on ℝN.Remark 1.
Suppose that∀x∈ℝN, ax=1, we have ϕx=ϕpx=xp−2x,p>1. Then this function satisfies hypotheses Ha and Hϕ. Other cases that satisfy Hypotheses Ha and Hϕ are when η<ax<M with η,M>0 and the function ϕ:x⟼axxp−2x is a monotone homeomorphism on ℝN. This is the case, for example, when function a is equal to one of the following functions:(11)x⟼1+11+xp2orx⟼p+p−xex,∀x∈ℝN,with p>1. Indeed, for these examples, the potentials functions Φ corresponding to ϕ are respectively the functions x⟼1/p1+1/1+xpxp and x⟼1+1/exxp, ∀x∈ℝN.HF: F:0,T×ℝN⟶ℝN is a Caratheodory mapping such thatHF1: F (t, .) is continuously differentiable, for a.e. t∈0,THF2: F.,0∈L10,THF3: for each R>0 there is some αR∈L10,T such that(12)∇Ft,x≤αRt,a.e.t∈0,T,∀x∈ℝNwithx≤R.
Our notion of solution of problem (1) is defined as follows:Definition 1.
By a solution of the differential system (1), we will understand a function u:0,T⟶ℝN of class C1Ω,ℝN with ϕu′ absolutely continuous, which satisfies the equality in (1) a.e. on 0,T.
Now let us start the variational approach. In this purpose, forε>0, let Kε:W1,p0,T,ℝN⟶ℝ be defined by(13)Kεu=∫0TΦu′s+εΦusds,∀u∈W1,p0,T,ℝN.Lemma 1.
K
ε is proper, convex, and lower semicontinuous.Proof.
By hypothesisHϕ, it follows(14)c1u′pp+εupp≤Kεu≤1+εc2+c3u′pp+εupp,∀u∈W1,p0,T,ℝN.
WhenceKε is proper. Also since Φ is convex, it follows that Kε is convex. Finally, because of the lower semicontinuity of the functional norm on Banach space, Kε is lower semicontinuous (in short, lsc).Lemma 2.
K
ε
∈
C
1
W
1
,
p
0
,
T
,
ℝ
N
,
ℝ and(15)K′εu,w=∫0Tϕu′t,w′tℝN+εϕut,wtℝNdt,∀u,w∈W1,p0,T,ℝN,where ℝN is the inner product in ℝN.Proof.
Let us consider the product spaceH=Πi=1NLqΩ,ℝ, 1/p+1/q=1, equipped with the norm(16)xH=∑i=1Nxiqq1/q,∀x=x1,x2,…,xn−1,xn∈H.
We definegg1,g2,…,gn−1,gN:W1,p0,T,ℝN⟶H such that(17)gu=ϕu′+εϕu,∀u∈W1,p0,T,ℝN.
Let us show thatg is bounded.
We have(18)giuqq=∫0Tϕiu′t+εϕiutqdt=∫0Tau′tu′tp−2ui′t+εaututp−2uitqdt≤Mq∫0Tu′tp−1q+εutp−1qdt=Mqu′pp+εupp=Mquεp≤Mqη1ump,for someη1>0.
It follows(19)guH≤MNη11/qump−1,∀u∈W1,pΩ.
Sog is bounded.
Let us show thatg is continuous.
Letunn≥1 be a sequence such that un⟶u in W1,pΩ. Then un⟶u in LpΩ and un′⇀u′ in LpΩ. Whence ϕiun′⟶ϕiu′ in LqΩ and ϕiun⟶ϕiu in LqΩ. We set(20)giun=ϕiun′+εϕiun.
By the previous arguments, the sequencegiun⟶giu in LqΩ. We infer that the sequence gun⟶gu in H. So g is continuous.
We consider the functional:(21)Qv=∫0Tϕu′t,v′t+εϕut,vtdt,for allv∈W1,pΩ.
Let us show that the linear operatorQ is continuous on W1,pΩ.
Using Hölder’s inequality, we obtain(22)Qv=∫0Tϕu′t,v′t+εϕut,vtdt≤∫0Tϕu′tqdt1/q∫0Tv′tpdt1/p+ε∫0Tϕutqdt1/q∫0Tvtpdt1/p≤M1+εump/qvm,∀u∈W1,pΩ.
SoQ is continuous.
Let us show thatKε is Frechet differentiable in u∈W1,pΩ and Kε′u=ϕu′+εϕu in the sense of (15).
Using Fubini’s inequality, for arbitraryv∈W1,pΩ, we obtain(23)Kεu+v−Kεu−∫0Tϕu′t,v′t+εϕut,vtdt=∫0T∫01ddsΦu′t+sv′t+εddsΦut+svtdsdt−∫0T∫01ϕu′t,v′t+εϕut,vtdsdt=∫0T∫01ϕu′t+sv′t,v′t+εϕut+svt,vtdsdt−∫0T∫01ϕu′t,v′t+εϕut,vtdsdt=∫0T∫01ϕu′t+sv′t−ϕu′t,v′t+εϕut+svt−ϕut,vtdsdt≤∫01∫0Tϕu′t+sv′t−ϕu′tq1/qdt∫0Tv′tp1/pdtds+ε∫01∫0Tϕut+svt−ϕutq1/qdt∫0Tvtp1/pdtds≤∫01ϕu′+sv′−ϕu′H+εϕu+sv−ϕuHdsvm≤∫01Lu+sv−LuH+εΛu+sv−ΛuHdsvm,with L:W1,pΩ⟶H and Lu=ϕu′; Λ:W1,pΩ⟶H and Λu=ϕu.
Arguing as in the proof of the continuity ofg and the fact it is bounded on W1,pΩ, we show that L and Λ are continuous and bounded on W1,pΩ. Moreover, using Lebesgue’s dominated comvergence theorem, we have(24)Kεu+v−Kεu−∫0Tϕu′t,v′t+εϕut,vtdtvm≤∫01Lu+sv−LuH+εΛu+sv−ΛuHds⟶0,asvm⟶0.
Therefore, (15) is proved.
We know thatKε′:W1,pΩ⟶W1,pΩ∗ where W1,pΩ∗ denotes the dual of W1,pΩ.
Let us show thatKε′ is continuous.
We have(25)Kε′u−Kε′v,w=∫0Tϕu′t+εϕut−ϕv′t−εϕvt,wdt=∫0Tgut−gvt,wdt.
Using Hölder’s inequality, we have(26)Kε′u−Kε′v,w≤kgu−gvHwm,∀u,v,w∈W1,pΩ.
Whence(27)Kε′u−Kε′vW1,pΩ∗≤kgu−gvH.
Thus, sinceg is continuous, we obtain the continuity of Kε′.
We introduce also the functional:J:W1,p0,T,ℝN⟶−∞,+∞ defined by(28)Ju=ju0,uT,∀u∈W1,p0,T,ℝN.
Recall thatj is proper, convex, and lsc. Then, J is also proper, convex, and lsc. Let us set(29)Δε=Kε+J.
SinceKε and J are proper, convex and lsc, it follows that Δε is proper, convex and lsc on W1,p0,T,ℝN. Assuming that hypotheses HF1,HF2 and HF3 on the Caratheodory function F hold, for each R>0, we obtain:(30)Ft,x≤RαRt+Ft,0,for a.et∈0,Twithx≤R,with αR∈L10,T. Equation (30) comes from inequality (12) and the estimation:(31)Ft,x=∫01ddsFt,sxds+Ft,0≤∫01∇Ft,sx,xℝNds+Ft,0≤x∫01∇Ft,sxds+Ft,0,equation (30) and the embedding W1,p0,T,ℝN⊂C0,T,ℝN allow us to introduce the functional: ΨF:W1,p0,T,ℝN⟶ℝ defined by(32)ΨFu=−∫0TFt,utdt+∫0TFt,0dt,∀u∈W1,p0,T,ℝN.Lemma 3.
If the hypothesesHF1,HF2, and HF3 hold, then(33)ΨF∈C1W1,pΩ,ℝN,ℝ,(34)ΨF′u,v=−∫0T∇Ft,u,vdt,∀u,v∈W1,p0,T,ℝN.Proof.
The proof is similar to the one of Theorem 2.6 of [2].
Considering the functional framework in Section2, we set X=W1,p0,T,ℝN,E=ΨF in (32), G=Δε in (29) and BF,ε=B:(35)BF,ε=ΨF+Δε.Proposition 2.
Let the Caratheodory functionF:0,T×ℝN⟶ℝ satisfies HF1,HF2, and HF3 and let u∈W1,p0,T,ℝN. If u is a critical point of the functional BF,ε defined by (35), in the sense (7), i.e.,(36)ΨF′u,w−u+Δεw−Δεu≥0,∀w∈W1,p0,T,ℝN,then u is a solution of problem (1). The converse implication is also true.Proof.
We suppose thatu is a critical point of BF,ε. In the inequality (36), we take w=u+sv. Then dividing by s and letting s⟶0+, we obtain(37)ΨF′u,v+Kε′u,v+J′u;v≥0,∀v∈W1,p0,T,ℝN,where J′u;v is the directional derivative of the convex function J at u in the direction v. From (28) and (37), it follows(38)ΨF′u,v+Kε′u,v+j′u0,uT;v0,vT≥0,∀v∈W1,p0,T,ℝN.
SinceC0∞0,T,ℝN⊂W1,p0,T,ℝN, using Hahn–Banach’s theorem (see Theorem I.1 of Brezis [3]), (38) implies(39)ΨF′u,v+Kε′u,v=0,∀v∈C0∞0,T,ℝN.
Using (15), (34), and (39), we obtain(40)∫0Tϕu′t,v′tℝNdt=∫0T−εϕut+∇Ft,ut,vtℝNdt,∀v∈C0∞0,T,ℝN.
From hypothesisHϕb and u∈W1,p0,T,ℝN, it follows that(41)ϕu,ϕu′∈Lq0,T,ℝN,with1p+1q=1.
Also,HF3 implies(42)∇F.,u∈L10,T,ℝN.
Equations (40) and (42) imply(43)ϕu′∈W1,10,T,ℝN,(44)−ϕu′t′=−εϕut+∇Ft,ut.
Withϕ being a homeomorphism, (43) ensures that u∈C10,T,ℝN. This together with (44) shows that u is the solution of the differential system (1). Furthermore, (38) and (44) yield(45)j′u0,uT;v0,vT≥ϕu′0,v0−ϕu′T,vT,∀v∈W1,p0,T,ℝN.
Thus,(46)j′u0,uT;x,y≥ϕu′0,x−ϕu′T,y,∀x,y∈ℝN,which, by a standard result from convex analysis (Theorem 23.2 of Rockafellar [8]), means that the boundary conditions in (1) are true.
Now let us show the converse implication.
By multiplying (1) with v∈W1,p0,T,ℝN and integrating by parts on 0,T, we obtain(47)∫0Tϕu′t,v′tℝN+εϕut,vtℝNdt+ϕu′0,v′0ℝN−ϕu′T,v′TℝN=∫0T∇Ft,ut,vtℝNdt.
Using inequalities (15) and (34) in (47), it follows(48)ΨF′u,v+Kε′u,v+ϕu′0,v′0ℝN−ϕu′T,v′TℝN=0.
By using Theorem 23.2 of Rockafellar [8], we obtain(49)ΨF′u,v+Kε′u,v+j′u0,uT;v0,vT≥0.
Using some previous arguments, we infer that(50)ΨF′u,w−u+Δεw−Δεu≥0,∀u∈W1,p0,T,ℝN.


## 4. Existence Results for Problem (1)

At first, let us introduce the constant(51)λ1=λ1p,j,ε=ε+infu′ppupp:u∈W1,p0,T,ℝN\0,u0,uT∈Dj,for ε≥0. If Dj=ℝN×ℝN, then λ1p,j,0=0 and if Dj=0;0, then λ1p,j,0>0. To obtain the existence result, we make the following hypothesis:Hλ1 λ1p,j,ε>0.Proposition 3.
IfHλ1 is true, then(52)21/puλ1≤u′pp+εupp1/p≤uλ1,∀u∈DJ,where J is defined by (28).Proof.
SinceDJ=u∈W1,p0,T,ℝN:u0,uT∈Dj, by (51) we have(53)λ1=λ1p,j,ε=ε+infu′ppupp:u∈W1,p0,T,ℝN\0,u0,uT∈Dj.
Whence,(54)λ1≤ε+u′ppupp∀u∈W1,p0,T,ℝN\0,u0,uT∈Dj.
Then(55)λ1upp≤εupp+u′pp.
It follows(56)u′pp+λ1upp≤2u′pp+εupp.
Whence(57)2−1/Puλ1≤u′pp+εupp1/p.
Furthermore, we have(58)ε+u′ppupp≤λ1+u′ppupp.
Then(59)u′pp+εupp≤u′pp+λ1upp.
So(60)u′pp+εupp1/p≤uλ1.
Inequalities (57) and (60) yield the result.
If the nonlinearityF lies asymptotically on the left of λ1, then problem (1) is solvable.Theorem 1.
AssumeHF1,HF2, HF3, and Hλ1. If(61)limx⟶∞supFt,xc1xp<λ1,uniformly for a.e.t∈0,T,the problem (1) has at least a solution.Proof.
Let us show that the functionalBF,ε in (35) is sequentially weakly lsc and coercive on the space W1,p0,T,ℝN,λ1.
Let us show thatΨF in (32) is sequentially weakly continuous.
Letu,v∈W1,p0,T,ℝN be such that u∞,v∞≤M, with some M>0. By HF3, there is an α2M∈L10,T such that(62)∇Ft,x≤α2M,for a.et∈0,T,∀x∈ℝNwithα2M≤2M.
We estimate(63)ΨFu−ΨFv=∫0TFt,u−Ft,v=∫0T∫01ddsFt,u+sv−uds=∫0T∫01∇Ft,u+sv−u,v−uℝNds≤∫0T∫01∇Ft,u+sv−udsu−v∞,and by (62), it follows(64)ΨFu−ΨFv≤∫0Tα2Mtdtu−v∞.
By the compactness of the embeddingW1,p0,T,ℝN⊂C0,T,ℝN and (64), it follows that ΨF is sequentially weakly continuous on W1,p0,T,ℝN. Then, by the weak lower semicontinuity of Δε in (29), BF,ε is sequentially weakly lower semicontinuous.
Furthermore, from (61), there are constant σ∈0,λ1 and ρ>0 such that(65)Ft,x≤λ1−σc1xp,for  a.et∈0,T,∀x∈ℝNwithx>ρ.
Then, (30) and (65) yield(66)Ft,x≤ραpt+Ft,0+λ1−σc1xp,for a.et∈0,T,∀x∈ℝN,which, by (32), give(67)ΨFu≥−γρ−λ1−σc1upp,∀u∈W1,p0,T,ℝN,with γρ=ρ∫0Tαptdt+2∫0TF0,tdt. Using (29), (67), and (53) and the Proposition 3, we estimate BF,ε on DJ as follows:(68)BF,εu=ΨFu+Δεu≥−γρ−λ1−σc1upp+c1u′pp+εupp+Ju≥−γρ−λ1−σc1u′pp+εuppλ1+c1u′pp+εupp+Ju=−γρ+σλ1c1u′pp+εupp+Ju≥−γρ+σ2λ1c1uλ1p+Ju,∀u∈DJ.
Sincej is convex and lsc, it is bounded from below by an affine functional. Therefore, by (28) there are constants k1,k2,k3≥0 such that(69)BF,εu≥−γρ+σ2λ1c1uλ1p−k1u0−k2uT−k3,∀u∈DJ.
From (69) and the continuity embedding of W1,p0,T,ℝN⊂C0,T,ℝN, we obtain(70)BF,εu≥−γρ+σ2λ1c1uλ1p−k¯1uλ1−k¯2uλ1−k3,∀u∈DJ.with some k¯1,k¯2≥0. Consequently,(71)BF,εu⟶+∞,asuλ1⟶∞,meaning that BF,ε is coercive on W1,p,.λ1.
Since the functionalBF,ε is sequentially weakly lsc and coercive, by a well-known result from calculus of variations, BF,ε is bounded from below and attains its infimum at some u∈W1,p0,T,ℝN. Then, by Proposition 1 and 3, u is a solution of problem (1).
Remarque 2. The problem (1) incorporates classical problems of Dirichlet, Neumann, Periodic, antiperiodic, and Sturm–Louiville (see Remark 3.9 of Jebelean [5]).


## 5. Example

We consider the following problem:(72)−1+11+u′tp2ϕpu′t′+ε1+11+u′tp−12ϕput=βtϕput,a.e onΩ=0,T1+11+u′0p2ϕpu′0,−1+11+u′Tp2ϕpuT∈∂ju0,uT,where ϕpx=xp−2x,∀x∈ℝN, is the well-known p−Laplacian operator. Let us set Φx=1/p1+1/1+xpxp. Whence ∇Φx=ϕx=1+1/1+xp2ϕpx. When, p>1, we see that ϕ is a monotone homeomorphism such that ϕ0=0. Moreover, Φ satisfies hypothesis Hϕb. Therefore, by Theorem 1, problem (72) has at least a solution.


## 6. Conclusion

In this article, using variational method which relies on Szulkin theory, we have established existence for second-order problems with multivalued boundary conditions. We give an example but more examples and applications can be given.



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