# Positive Solutions of Sturm-Liouville Boundary Value Problems in Presence of Upper and Lower Solutions

Authors: Li Zhang; Xiankai Huang; Weigao Ge
Journal: International Journal of Differential Equations
 (2011)

---

## Abstract

We consider a kind of Sturm-Liouville boundary value problems. Using variational techniques combined with the methods of upper-lower solutions, the existence of at least one positive solution is established. Moreover, the upper solution and the lower solution are presented.


---

## Body

## 1. Introduction

The Sturm-Liouville boundary value problems (for short, BVPs) have received a lot of attention. Many works have been carried out to discuss the existence of at least one solution or multiple solutions. The methods used therein mainly depend on the Leray-Schauder continuation theorem and the Mawhin continuation theorem. Since it is very difficult to give the corresponding Euler functional for Sturm-Liouville BVPs and verify the existence of critical points for the Euler functional, few people consider the existence of solutions for Sturm-Liouville BVPs by critical point theory and many works considered the existence of solutions for Dirichlet BVPs. For example, by a three-critical-point theorem due to Ricceri [1], Bonanno [2] considered Dirichlet problems. Moreover, Afrouzi and Heidarkhani [3] also considered the existence of three solutions for a kind of Dirichlet BVP. By using an appropriate variational framework, the authors [4] considered the existence of positive solutions for the Dirichlet BVP.In this paper, using variational methods combined with the methods of upper-lower solutions, we consider the positive solutions of the following BVP:(1.1)-(ϕp(x′(t)))′=-a(t)ϕp(x)+f(t,x),t∈[0,1],α1x(0)-α2x′(0)=0,β1x(1)+β2x′(1)=0,
where p>1, ϕp(x)=|x|p-2x, α1,α2,β1,β2≥0, α12+α22>0, β12+β22>0.The paper is organized as follows. In the forthcoming section, we give the Euler functional of BVP(1.1) and some basic lemmas. In Section 3, firstly, we give an upper solution of BVP(1.1), then, by the mountain pass lemma, the lower solution of BVP(1.1) is obtained. At last, we show the existence of at least one positive solution of BVP(1.1) based on the upper solution and the lower solution we obtain.


## 2. Preliminary

The Sobolev spaceW1,p[0,1] is defined by(2.1)W1,p[0,1]={x:[0,1]⟶R∣xisabsolutelycontinuousandx′∈Lp(0,1;R)}
and is endowed with the norm(2.2)‖x‖=(∫01|x(t)|pdt+∫01|x′(t)|pdt)1/p.
Then, W1,p[0,1] is a separable and reflexive Banach space [5].Lemma 2.1 (see [6]).
There exists a positive constantcp such that
(2.3)(|x|p-2x-|y|p-2y,x-y)≥{cp|x-y|p,p≥2,cp|x-y|2(|x|+|y|)2-p,1<p<2
for any x,y∈RN. Here (x,y)=x·yT.
Forx∈C[0,1], suppose that ∥x∥∞=maxt∈[0,1]|x(t)|, ∥x∥m=mint∈[0,1]|x(t)|.Lemma 2.2 (see [7]).
Ifx∈W1,p[0,1], then, ∥x∥∞≤2∥x∥.Lemma 2.3 (see [8]).
Forx∈X, let x±=max{±x,0}; then, the following properties hold:(i)
x∈X⇒x+,x-∈X;(ii)
x=x+-x-;(iii)
∥x+∥X≤∥x∥X;(iv)
if(xn)n∈N uniformly converges to x in C([0,1]), then, (xn+)n∈N uniformly converges to x+;(v)
ϕp(x)x+=|x+|p, ϕp(x)x-=-|x-|p.In the following, we state the (C ) condition [9].
(C)
Every sequence(xn)n∈N⊂H such that the following conditions hold:(i)
(φ(xn))n∈N is bounded;(ii)
(1+∥xn∥H)∥φ′(xn)∥H*→0, n→∞has a subsequence which converges strongly in H.With a similar proof of Lemma 2.5 [8], one has the following lemma.Lemma 2.4.
Ifx(t)∈W1,p[0,1] is a critical point of the Euler functional
(2.4)φ(x)=1p∫01a(t)|x|pdt+1p∫01|x′|pdt-∫01F(t,x)dt+α2pα1|α1x(0)α2|p+β2pβ1|β1x(1)β2|p,
then, x(t) is a solution of BVP (1.1). Here, F(t,x)=∫0xf(t,s)ds.Remark 2.5.
Whileα2=0, the Euler functional φ(x) does not include (1/p)(α1/α2)p-1|x(0)|p, while β2=0, φ(x) does not include (1/p)(β1/β2)p-1|x(1)|p. Hence, in order to be convenient, we assume that α1,α2,β1,β2>0.With little modification to the proof of Theorem 1.4 in [7], we obtain the following.Remark 2.6.
φ is continuously differentiable on W1,p[0,1], and, by computation, one has
(2.5)〈φ′(x),y〉=∫01a(t)ϕp(x)ydt+∫01ϕp(x′)y′dt-∫01f(t,x)ydt+ϕp(α1x(0)α2)y(0)+ϕp(β1x(1)β2)y(1),x,y∈W1,p[0,1].Definition 2.7.
u∈W1,p[0,1] is an upper solution of BVP (1.1) if it satisfies
(2.6)-(ϕp(u′(t)))′+a(t)ϕp(u)-f(t,u)≥0,t∈[0,1],α1u(0)-α2u′(0)≥0,β1u(1)+β2u′(1)≥0.
If u is not a solution of BVP(1.1), then, u is a strict upper solution.Definition 2.8.
v∈W1,p[0,1] is a lower solution of BVP(1.1) if it satisfies
(2.7)-(ϕp(v′(t)))′+a(t)ϕp(v)-f(t,v)≤0,t∈[0,1],α1v(0)-α2v′(0)≤0,β1v(1)+β2v′(1)≤0.
If v is not a solution of BVP(1.1), then, v is a strict lower solution.Definition 2.9.
x∈W1,p[0,1] is said to be a positive solution of BVP(1.1) if x(t)≥0, x(t)≢0, t∈[0,1].


## 3. Existence of Positive Solutions

Choosex0∈W1,p[0,1] and x0(t)>0, t∈[0,1] satisfying -(ϕp(x0′))′=1, then, x0(t)=c2+∫0tϕq(-s+c1)ds where (1/p)+(1/q)=1,c1,c2 are constants. If we choose c1≥1, c2≥(α2ϕq(c1))/α1, x0(t) satisfies α1x0(0)-α2x0′(0)≥0, β1x0(1)+β2x0′(1)≥0. Moreover, x0′(t)=ϕq(-t+c1) is continuous.Lemma 3.1.
Assume(A1)
f(t,x)∈C([0,1]×[0,+∞)),lim¯x→+∞(f(t,x)/ϕp(x))<a(t),t∈[0,1],is satisfied; then, x¯=a01/(p-1)x0 is a strict upper solution of BVP (1.1). Here a0>1 is some positive constant.Proof.
From (A1), there exists a constant N>0 such that
(3.1)f(t,x)ϕp(x)<a(t),x>N.
Hence,
(3.2)f(t,x)<a(t)ϕp(x)+a0,t∈[0,1],
holds for x≥0 and some large positive constant a0>1. Then,
(3.3)f(t,a01/(p-1)x0)<a(t)ϕp(a01/(p-1)x0)+a0=a(t)ϕp(a01/(p-1)x0)-(ϕp(a01/(p-1)x0′))′,t∈[0,1],
that is, -(ϕp(x¯′))′+a(t)ϕp(x¯)-f(t,x¯)>0, t∈[0,1]. Obviously, α1x¯(0)-α2x¯′(0)≥0, β1x¯(1)+β2x¯′(1)≥0. Therefore, from Definition 2.7, one has that x¯=a01/(p-1)x0 is a strict upper solution of BVP (1.1).In the following, we assume the following conditions.(A2)
There existδ>0 and g(x):[0,+∞)→[0,+∞) satisfying g(mx)≥mp-1g(x) for m<1, g(0)≠0, f(t,x)>g(x) for x∈(0,δ], t∈[0,1].(A3)
There existsμ>p such that μG(x)≤g(x)x, x≥0, G(x)=∫0xg(s)ds.Consider the auxiliary BVP(3.4)-(ϕp(x′(t)))′=-a(t)ϕp(x)+g(x+),t∈[0,1],α1x(0)-α2x′(0)=0,β1x(1)+β2x′(1)=0.
Obviously, the corresponding Euler functional of BVP(3.4) is(3.5)φ+(x)=1p∫01a(t)|x|pdt+1p∫01|x′|pdt-∫01(G(x+(t))-g(0)x-)dt+α2pα1|α1x(0)α2|p+β2pβ1|β1x(1)β2|p.
Obviously, φ+ is continuously differentiable on W1,p[0,1], and, by computation, one has(3.6)〈φ′(x),y〉=∫01a(t)ϕp(x)ydt+∫01ϕp(x′)y′dt-∫01g(x+)ydt+ϕp(α1x(0)α2)y(0)+ϕp(β1x(1)β2)y(1),x,y∈W1,p[0,1].Lemma 3.2.
Ifx(t)∈W1,p[0,1] is a solution of BVP (3.4), then, x(t)≥0.Proof.
Letx(t)∈W1,p[0,1] be a solution of the BVP (3.4). If there exists a subset E0⊂[0,1], measE0≠0, x(t)≡0 for t∈E0, then from the BVP (3.4), one has g(0)≡0 for t∈E0 which contradicts with the assumptions. Moreover, x- is an absolutely continuous function on [0,1], and so the fundamental theorem of calculus ensures the existence of a set E1⊂[0,1] such that meas([0,1]∖E1)=0 and x- is differentiable on E1, (x-)′∈L1[0,1],
(3.7)0=∫01((ϕp(x′(t)))′-a(t)ϕp(x(t))+g(x+))x-dt≥x-(1)ϕp(x′(1))-x-(0)ϕp(x′(0))-∫E1ϕp(x′(t))(x-)′dt-∫01a(t)ϕp(x(t))x-dt≥x-(1)ϕp(x′(1))-x-(0)ϕp(x′(0))+∫E1|(x-)′|pdt+∫01a(t)|x-|pdt≥x-(1)ϕp(-β1x(1)β2)-x-(0)ϕp(α1x(0)α2)+min⁡{‖a‖m,1}‖x-‖p≥min⁡{‖a‖m,1}‖x-‖p.
Therefore, for a.e. t∈[0,1], x-=0. Since x(t) is absolutely continuous on [0,1], then, x(t)≥0 for t∈[0,1].Lemma 3.3.
Assume that (A2), (A3) hold; then, BVP(3.4) has a solution x1, that is, BVP(3.4) has a positive solution x1.Proof.
Assume that(xn)n∈N⊂W1,p[0,1] satisfies (i) and (ii) of the (C) condition; then,
(3.8)|φ+(xn)|≤c1,‖φ+′(xn)‖(1+‖xn‖)≤εn.
Here, c1 is some positive constant and εn→0, n→∞.
First, we show that(xn-)n≥1⊂W1,p[0,1] is bounded. Indeed, from (3.8), we have
(3.9)|〈φ+′(xn),u〉|≤εn,u∈W1,p[0,1].
Choose u=-xn-; then,
(3.10)|〈φ+′(xn),-xn-〉|=∫01a(t)|xn-|pdt+1p∫01|(xn-)′|pdt+ϕp(α1α2)|xn-(0)|p+ϕp(β1β2)|xn-(1)|p+∫01g(xn+)xn-dt.
Hence, (xn-)n∈N is bounded.
Moreover,(3.11)〈φ+′(xn),xn+〉=∫01a(t)ϕp(xn)xn+dt+∫01ϕp(xn′)(xn+)′dt-∫01g(xn+)xn+dt+ϕp(α1xn(0)α2)xn+(0)+ϕp(β1xn(1)β2)xn+(1)=∫01a(t)|xn+|pdt+∫01|(xn+)′|pdt-∫01g(xn+)xn+dt+ϕp(α1α2)|xn+(0)|p+ϕp(β1β2)|xn+(0)|p.
For large n,
(3.12)(μ+1)c1=c1+μc1≥μφ+(xn)-〈φ+′(xn),xn+〉=μp∫01a(t)|xn|pdt+μp∫01|xn′|pdt-μ∫01G(xn+(t))dt+μ∫01g(0)xn-(t)dt+μα2pα1|α1xn(0)α2|p+μβ2pβ1|β1xn(1)β2|p-∫01a(t)|xn+|pdt-∫01|(xn+)′|pdt+∫01g(xn+)xn+(t)dt-ϕp(α1α2)|xn+(0)|p-ϕp(β1β2)|xn+(1)|p≥μp∫01a(t)|xn|pdt+μp∫01|xn′|pdt-∫01a(t)|xn+|pdt-∫01|(xn+)′|pdt≥(μp-1)∫01a(t)|xn+|pdt+(μp-1)∫01|(xn+)′|pdt≥(μp-1)min⁡{‖a‖m,1}‖xn+‖p.
Hence, (xn+)n∈N is bounded; then, (xn)n∈N is uniformly bounded in W1,p[0,1]. By the compactness of the embedding W1,p[0,1]C[0,1], the sequence (xn)n∈N has a subsequence, again denoted by (xn)n∈N for convenience, such that
(3.13)xn⇀xweaklyinW1,p[0,1],xn⟶xstronglyinC[0,1].
Moreover,
(3.14)〈φ+′(xn)-φ+′(xm),xn-xm〉=∫01(ϕp(xn′)-ϕp(xm′))(xn′-xm′)dt+∫01a(t)(ϕp(xn)-ϕp(xm))(xn-xm)dt-∫01(g(xn+)-g(xm+))(xn-xm)dt+(ϕp(α1xn(0)α2)-ϕp(α1xm(0)α2))(xn(0)-xm(0))+(ϕp(β1xn(1)β2)-ϕp(β1xm(1)β2))(xn(1)-xm(1)).
Since xn(t)→x(t) in C[0,1], then, (ϕp(α1xn(0)/α2)-ϕp(α1xm(0)/α2))(xn(0)-xm(0))→0, (ϕp(β1xn(1)/β2)-ϕp(β1xm(1)/β2))(xn(1)-xm(1))→0, (g(xn+)-g(xm+))(xn-xm)→0, ∫01(xn(t)-xm(t))dt→0, n,m→∞. Moreover,
(3.15)|∫01a(t)(ϕp(xn)-ϕp(xm))(xn-xm)dt|≤‖a‖∞‖xn-xm‖∞∫01(ϕp(xn)-ϕp(xm))dt⟶0,asn,m⟶∞.
From|〈φ′(xn)-φ′(xm),xn-xm〉|≤(∥φ′(xn)∥+∥φ′(xm)∥)·(∥xn∥+∥xm∥) and ∥xn∥+∥xm∥ is bounded in W1,p[0,1], ∥φ′(xn)∥→0, ∥φ′(xm)∥→0, m,n→∞, and one has 〈φ′(xn)-φ′(xm),xn-xm〉→0. Hence,
(3.16)∫01(ϕp(xn′)-ϕp(xm′))(xn′-xm′)dt⟶0,n,m⟶∞.
Ifp≥2, from Lemma 2.1, there exists a positive constant cp such that
(3.17)∫01(ϕp(xn′)-ϕp(xm′))(xn′-xm′)dt≥cp∫01|xn′-xm′|pdt.
Ifp<2, by Lemma 2.1, the Hölder inequality, and the boundedness of (xn)n∈N in W1,p[0,1], one has
(3.18)∫01|xn′-xm′|pdt=∫01|xn′-xm′|p(|xn′|+|xm′|)p(2-p)/2(|xn′|+|xm′|)p(2-p)/2dt≤(∫01|xn′-xm′|2(|xn′|+|xm′|)2-pdt)p/2(∫01(|xn′|+|xm′|)pdt)(2-p)/2≤cp-p/2(∫01(ϕp(xn′)-ϕp(xm′))(xn′-xm′)dt)p/2×2(p-1)(2-p)/2(∫01(|xn′|p+|xm′|p)dt)(2-p)/2≤cp-p/2(∫01(ϕp(xn′)-ϕp(xm′))(xn′-xm′)dt)p/22((p-1)(2-p))/2×(‖xn‖p+‖xm‖p)(2-p)/2.
From (3.17) and (3.18), we have ∫01|xn′-xm′|pdt→0 as n,m→∞. Then, ∥xn-xm∥→0, that is, (xn)n∈N is a Cauchy sequence in W1,p[0,1]. By the completeness of W1,p[0,1], we have xn→x in W1,p[0,1]. From the discussion above, φ(x) satisfies the (C) condition.
Fort>0, x>0, one has
(3.19)d(G(t-1x)tμ)dt=tμ-1(μG(t-1x)-t-1xg(t-1x))≤0,
that is, G(t-1x)tμ is nonincreasing in t. Assume that M=maxx∈[0,1]G(x),
(3.20)G(x+)≤G(x+|x|)|x|μ≤M|x|μ,0<|x|≤1.
Hence,
(3.21)φ+(x)≥1p∫01a(t)|x|pdt+1p∫01|x′|pdt-∫01G(x+)dt≥1pmin⁡{‖a‖m,1}‖x‖p-M‖x‖μ.
Obviously, there exists ρ>0 such that, for ∥x∥=ρ, (1/p)min{|a|m,1}ρp-Mρμ=α>0.
On the other hand,G(x)≥G(1)xμ for x>1; then, by Lemma 2.2(3.22)φ+(x)≤1pmax⁡{∥a∥∞,1}‖x‖p-G(1)∫01|x|μdt+∫01g(0)x-dt+2p+1p(ϕp(α1α2)+ϕp(β1β2))‖x‖p.
Let e be some large positive constant. Since μ>p, φ+(e)<0. Moreover, φ+(0)=0. From the mountain pass lemma [10], φ+ possesses a critical value c≥α, that is, there exists x1 such that φ+′(x1)=0, φ+(x1)=c≥α>0. Then, from Lemma 2.4, one has that BVP(3.4) has a positive solution x1 and x1≢0, t∈[0,1].Lemma 3.4.
Assume that (A2), (A3) hold; then, BVP(1.1) has a strict lower solution x̲=βx1 where β is some positive constant and x1 is the positive solution of BVP(3.4) one obtains that in Lemma 3.3.Proof.
Assumeβ∈(0,1] is small enough such that βx1∈(0,δ] and x¯(t)-βx1(t)≥0, x¯(t)-βx1(t)≢0, t∈[0,1]. Then,
(3.23)-(ϕp(x̲′))′=-βp-1(ϕp(x1′))′=-βp-1a(t)ϕp(x1)+βp-1g(x1)≤-a(t)ϕp(x̲)+g(x̲)<-a(t)ϕp(x̲)+f(t,x̲).
Moreover, α1x̲(0)-α2x̲′(0)=0, β1x̲(1)+β2x̲′(1)=0. Hence, x̲ is a strictly lower solution of BVP(1.1) and x̲≤x¯, x̲≢x¯, t∈[0,1].Theorem 3.5.
Assume that (A1)–(A3) hold; then, BVP(1.1) has a positive solution x* and x̲≤x*≤x¯.Proof.
LetI=[x̲,x¯]={x∈W1,p[0,1]∣x̲≤x≤x¯}. Make a truncation function of f(t,x) as
(3.24)f¯(t,x)={f(t,x¯),x>x¯,f(t,x),x̲≤x≤x¯,f(t,x̲),x<x̲,
and assume that F¯(t,x)=∫0xf¯(t,s)ds. Consider the following BVP:
(3.25)-(ϕp(x′(t)))′=-a(t)ϕp(x)+f¯(t,x),t∈[0,1],α1x(0)-α2x′(0)=0,β1x(1)+β2x′(1)=0.
The corresponding Euler functional of BVP(3.25) is
(3.26)φ¯(x)=1p∫01a(t)|x|pdt+1p∫01|x′|pdt-∫01F¯(t,x)dt+α2pα1|α1x(0)α2|p+β2pβ1|β1x(1)β2|p.
It is obvious that φ¯(x) is weakly lower semicontinuous. Since x¯ and x̲ are continuous on [0,1], F¯(t,x) is continuous, φ¯(x) is coercive. Hence, φ¯(x) can attain its infimum in W1,p[0,1]. Without loss of generality, we may assume that φ¯(x) attains its infimum in x*. In the following, we show that x* is a solution of BVP(1.1).
Assume thatx*-x̲ has a negative minimum, and let t0=sup{t∈[0,1]∣(x*-x̲)(t)=mins∈[0,1]((x*-x̲)(s))}.
Ift0=0, then,
(3.27)0≤x*(0)′-x̲(0)′≤α1α2(x*(0)-x̲(0))<0,
which reaches a contradiction. Similarly, t0≠1.
Ift0∈(0,1), there exist an open interval I0 and t1∈I0 with t1<t0, x*(t)<x̲(t), t∈I0, x*(t1)′<x̲(t1)′. Hence,
(3.28)0>ϕp((x*)′(t1))-ϕp(x̲′(t1))=∫t0t1[(ϕp((x*)′(s)))′-(ϕp(x̲′(s)))′]ds≥∫t1t0[-a(s)ϕp(x*(s))+f¯(s,x*(s))+a(s)ϕp(x̲(s))-f(s,x̲(s))]ds=∫t1t0a(s)[ϕp(x̲(s))-ϕp(x*(s))]ds>0.
From the discussion above, one has x*(t)≥x̲(t). Similarly, x*(t)≤x¯(t). Then, x*∈I. Since x̲(t) and x¯(t) are the strictly lower and upper solutions of BVP(1.1), respectively, x*(t)x¯(t), x*(t)x̲(t). Therefore, we obtain a positive solution of BVP (1.1).



---

*Source: 383294-2011-10-30.xml*