# Multiple Solutions for a Class of Multipoint Boundary Value Systems Driven by a One-Dimensional(p1,…,pn)-Laplacian Operator

Authors: Shapour Heidarkhani
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

Employing a recent three critical points theorem due to Bonanno and Marano (2010), the existence of at least three solutions for the following multipoint boundary value system-(|ui′|pi-2ui′)′=λFui(x,u1,…,un) in (0,1), ui(0)=∑j=1majui(xj), ui(1)=∑j=1mbjui(xj) for 1≤i≤n, is established.


---

## Body

## 1. Introduction

In this work, we consider the following multipoint boundary value system(1.1)-(|ui′|pi-2ui′)′=λFui(x,u1,…,un)in(0,1),ui(0)=∑j=1majui(xj),ui(1)=∑j=1mbjui(xj),
for 1≤i≤n, where pi>1 for 1≤i≤n, λ>0, m,n≥1, F:[0,1]×ℝn→ℝ is a function such that F(·,t1,…,tn) is continuous in [0,1] for all (t1,…,tn)∈ℝn, F(x,·,…,·) is C1 in ℝn for every x∈[0,1] and F(x,0,…,0)=0 for all x∈[0,1], aj,bj∈ℝ for j=1,…,m and 0<x1<x2<x3<⋯<xm<1, and Fui denotes the partial derivative of F with respect to ui for 1≤i≤n.The study of multiplicity of solutions is an important mathematical subject which is also interesting from the practical point of view because the physical processes described by boundary value problems for differential equations exhibit, generally, more than one solution. In [1–3], Ricceri proposed and developed an innovative minimal method for the study of nonlinear eigenvalue problems. Following that, Bonanno [4] gave an application of the method to the two-point problem
(1.2)u″+λf(u)=0in(0,1),u(0)=u(1)=0.
Bonanno also gave more precise versions of the three critical points of Ricceri in [5, 6]. In particular, in [5], an upper bound of the interval of parameters λ for which the functional has three critical points is established. Candito [7] extended the main result of [4] to the nonautonomous case
(1.3)u″+λf(x,u)=0in(a,b),u(a)=u(b)=0.
In [8], He and Ge extended the main results of [4, 7] to the quasilinear differential equation
(1.4)(φp(u′))′+λf(x,u)=0in(a,b),u(a)=u(b)=0.
In [9], the authors extended the main results of [4, 7, 9] to the quasilinear differential equation with Sturm-Liouville boundary conditions
(1.5)(|u′|p-2u′)′+λf(x,u)=0in(a,b),α1u(a)-α2u′(a)=0,β1u(b)-β2u′(b)=0,
where p>1 is a constant, λ is a positive parameter, a,b∈ℝ;a<b. In particular, in [10], the authors motivated by these works, established some criteria for the existence of three classical solutions of the system (1.1), while in [11], based on Ricceri’s three critical points theorem [3], the existence of at least three classical solutions to doubly eigenvalue multipoint boundary value systems was established.In the present paper, based on a three critical points theorem due to Bonanno and Marano [12], we ensure the existence of least three classical solutions for the system (1.1).Several results are known concerning the existence of multiple solutions for multipoint boundary value problems, and we refer the reader to the papers [13–16] and the references cited therein.Here and in the sequel,X will denote the Cartesian product of n space
(1.6)Xi={ξ∈W1,pi([0,1]);ξ(0)=∑j=1majξ(xj),ξ(1)=∑j=1mbjξ(xj)},
for i=1,…,n, that is, X=X1×⋯×Xn equipped with the norm
(1.7)‖(u1,…,un)‖=∑i=1n‖ui′‖pi,
where
(1.8)‖ui′‖pi=(∫01|ui′(x)|pidx)1/pi
for 1≤i≤n.We say thatu=(u1,…,un) is a weak solution to (1.1) if u=(u1,…,un)∈X and(1.9)∫01∑i=1n|ui′(x)|pi-2ui′(x)vi′(x)dx-λ∫01∑i=1nFui(x,u1(x),…,un(x))vi(x)dx=0,for every(v1,…,vn)∈X.A special case of our main result is the following theorem.Theorem 1.1.
Letf,g:ℝ2→ℝ be two positive continuous functions such that the differential 1-form  w:=f(ξ,η)dξ+g(ξ,η)dη is integrable, and let  F be a primitive of  w such that  F(0,0)=0. Fix  p,q>2,  0<x1<x2<1, and assume that
(1.10)lim⁡inf⁡(ξ,η)→(0,0)F(ξ,η)|ξ|p/p+|η|q/q=limsup⁡|ξ|→+∞,|η|→+∞F(ξ,η)|ξ|p/p+|η|q/q=0,
then there is λ*>0 such that for each λ>λ*, the problem
(1.11)-(|u1′|p-2u1′)′=λf(u1,u2)in(0,1),-(|u2′|q-2u2′)′=λg(u1,u2)in(0,1),ui(0)=a1ui(x1)+a2ui(x2),ui(1)=b1ui(x1)+b2ui(x2)
admits at least two positive classical solutions.The main aim of the present paper is to obtain further applications of [12, Theorem 2.6] (see Theorem 2.1 in the next section) to the system (1.1), and the obtained results are strictly comparable with those of [9–11], and here we wil give the exact collocation of the interval of positive parameters.For other basic notations and definitions, we refer the reader to [17–26]. We note that some of the ideas used here were motivated by corresponding ones in [10].


## 2. Main Results

Our main tool is a three critical points theorem obtained in [12] (see also [1, 2, 5, 27] for related results), which is a more precise version of Theorem 3.2 of [28], to transfer the existence of three solutions of the system (1.1) into the existence of critical points of the Euler functional. We recall it here in a convenient form (see [23]).Theorem 2.1 (see [12, Theorem 2.6]).
LetX be a reflexive real Banach space,  Φ:X→ℝ be a coercive continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on  X*,  Ψ:X→ℝ be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that  Φ(0)=Ψ(0)=0. Assume that there exist r>0 and x¯∈X, with r<Φ(x¯) such that(κ1)
supΦ(x)≤rΨ(x)/r<Ψ(x¯)/Φ(x¯),(κ2)
for eachλ∈Λr:=]Φ(x¯)/Ψ(x¯),r/supΦ(x)≤rΨ(x)[, the functional Φ-λΨ is coercive.
then, for eachλ∈Λr, the functional Φ-λΨ has at least three distinct critical points in X.Put(2.1)k=max⁡{sup⁡ui∈Xi∖{0}max⁡x∈[0,1]⁡|ui(x)|pi‖ui‖pipi;for1≤i≤n}.
Since pi>1 for 1≤i≤n, and the embedding X=X1×⋯×Xn↪(C0([0,1]))n is compact, one has k<+∞. Moreover, from [10, Lemma3.1], one has
(2.2)sup⁡u∈Xi∖{0}max⁡x∈[0,1]⁡|u(x)|‖u′‖pi≤12(1+∑j=1m|aj||1-∑j=1maj|+∑j=1m|bj||1-∑j=1mbj|).Putϕpi(s)=|s|pi-1s for 1≤i≤n. Let ϕpi-1 denotes the inverse of ϕpi for 1≤i≤n. then, ϕpi-1(t)=ϕqi(t) where 1/pi+1/qi=1. It is clear that ϕpi is increasing on ℝ,
(2.3)lim⁡t→-∞⁡ϕpi(t)=-∞,lim⁡t→+∞ϕpi(t)=+∞.Lemma 2.2 (see [10, Lemma 3.3]).
For fixedλ∈ℝ and  u=(u1,…,un)∈(C([0,1]))n, define  αi(t;u):ℝ→ℝ by
(2.4)αi(t;u)=∫01ϕpi-1(t-λ∫0δFui(ξ,u1(ξ),…,un(ξ))dξ)dδ+∑j=1majui(xj)-∑j=1mbjui(xj).
then the equation
(2.5)αi(t;u)=0has a unique solution tu,i.Direct computations show the following.Lemma 2.3 (see [10, Lemma3.4]).
The functionu=(u1,…,un) is a solution of the system (1.1) if and only if  ui(x) is a solution of the equation
(2.6)ui(x)=∑j=1majui(xj)+∫0xϕpi-1(tu,i-λ∫0δFui(ξ,u1(ξ),…,un(ξ))dξ)dδ,
for 1≤i≤n, where tu,i is the unique solution of (2.5).Lemma 2.4.
A weak solution to the systems (1.1) coincides with classical solution one. Proof.
Suppose thatu=(u1,…,un)∈X is a weak solution to (1.1), so
(2.7)∫01∑i=1nϕpi(ui′(x))vi′(x)dx-λ∫01∑i=1nFui(x,u1(x),…,un(x))vi(x)dx,
for every (v1,…,vn)∈X. Note that, in one dimension, any weakly differentiable function is absolutely continuous, so that its classical derivative exists almost everywhere, and that the classical derivative coincides with the weak derivative. Now, using integration by part, from (2.7), we obtain
(2.8)∑i=1n∫01[(ϕpi(ui′(x)))′+λFui(x,u1(x),…,un(x))]vi(x)dx=0,
and so for 1≤i≤n,
(2.9)(ϕpi(ui′(x)))′+λFui(x,u1(x),…,un(x))=0,
for almost every x∈(0,1). Then, by Lemmas 2.2 and 2.3, we observe
(2.10)ui(x)=∑j=1majui(xj)+∫0xϕpi-1(tu,i-λ∫0δFui(s,u1(s),…,un(s))ds)dδ,
for 1≤i≤n, where tu,i is the unique solution of (2.5). Hence, ui∈C1([0,1]) and ϕpi(ui′(x))∈C1([0,1]) for 1≤i≤n, namely u=(u1,…,un) is a classical solution to the system (1.1).For allγ>0, we denote by K(γ) the set (2.11){(t1,…,tn)∈Rn:∑i=1n|ti|pipi≤γ}.
Now, we formulate our main result as follows.Theorem 2.5.
Assume that there exist2m constants  aj,bj for  1≤j≤m with  ∑j=1maj≠1 and  ∑j=1mbj≠1, a positive constant  r and a function  w=(w1,…,wn)∈X such that (A1)
∑i=1n(∥wi′∥pipi/pi)>r,(A2)
∫01sup(t1,…,tn)∈K(kr)F(x,t1,…,tn)dx<(r∏i=1npi)∫01F(x,w1(x),…,wn(x))dx/∑i=1n∏j=1,j≠inpj∥wi′∥pipiwhere K(kr)={(t1,…,tn)|∑i=1n(|ti|pi/pi)≤kr}(see (2.11)),(A3)
limsup|t1|→+∞,…,|tn|→+∞F(x,t1,…,tn)/∑i=1n(|ti|pi/pi)<∫01sup(t1,…,tn)∈K(kr)×F(x,t1,…,tn)dx/kr uniformly with respect to x∈[0,1].Then, for each λ∈Λr:=]∑i=1n(∥wi′pipi∥/pi)/∫01F(x,w1(x),…,wn(x))dx,r/∫11sup(t1,…,tn)∈K(kr)F(x,t1,…,tn)dx[, the system (1.1) admits at least three distinct classical solutions in X.Proof.
In order to apply Theorem2.1 to our problem, we introduce the functionals Φ,Ψ:X→ℝ for each u=(u1,…,un)∈X, as follows:
(2.12)Φ(u)=∑i=1n‖ui′‖pipipi,Ψ(u)=∫01F(x,u1(x),…,un(x))dx.
Since pi>1 for 1≤i≤n, X is compactly embedded in (C0([0,1]))n and it is well known that Φ and Ψ are well defined and continuously differentiable functionals whose derivatives at the point u=(u1,…,un)∈X are the functionals Φ′(u), Ψ′(u)∈X*, given by
(2.13)Φ′(u)(v)=∫01∑i=1n|ui′(x)|pi-2ui′(x)vi′(x)dx,Ψ′(u)(v)=∫01∑i=1nFui(x,u1(x),…,un(x))vi(x)dx
for every v=(v1,…,vn)∈X, respectively, as well as Ψ is sequentially weakly upper semicontinuous. Furthermore, Lemma 2.6 of [11] gives that Φ′ admits a continuous inverse on X*, and since Φ′ is monotone, we obtain that Φ is sequentially weakly lower semiacontinuous (see [29, Proposition 25.20]). Moreover, Ψ′:X→X* is a compact operator. From assumption (A1), we get 0<r<Φ(w). Since from (2.1) for each ui∈Xi,
(2.14)sup⁡x∈[0,1]|ui(x)|pi≤k‖ui′‖pi
for i=1,…,n, we have
(2.15)sup⁡x∈[0,1]∑i=1n|ui(x)|pipi≤k∑i=1n‖ui′‖pipipi,
for each u=(u1,…,un)∈X, and so using (2.15), we observe
(2.16)Φ-1(]-∞,r])={(u1,…,un)∈X;Φ(u1,…,un)≤r}={(u1,…,un)∈X;∑i=1n‖ui′‖pipipi≤r}⊆{(u1,…,un)∈X;∑i=1n|ui(x)|pipi≤kr∀x∈[0,1]},
and it follows that
(2.17)sup⁡(u1,…,un)∈Φ-1(]-∞,r])Ψ(u1,…,un)=sup⁡(u1,…,un)∈Φ-1(]-∞,r])∫01F(x,u1(x),…,un(x))dx≤∫01sup⁡(t1,…,tn)∈K(kr)F(x,t1,…,tn)dx.
Therefore, owing to assumption (A2), we have
(2.18)sup⁡u∈Φ-1(]-∞,r])Ψ(u1,…,un)=sup⁡(u1,…,un)∈Φ-1(]-∞,r])∫01F(x,u1(x),…,un(x))dx≤∫01sup⁡(t1,…,tn)∈K(kr)F(x,t1,…,tn)dx<(r∏i=1npi)∫01F(x,w1(x),…,wn(x))dx∑i=1n∏j=1,j≠inpj‖wi′‖pipi<r∫01F(x,w1(x),…,wn(x))dx∑i=1n(‖wi′‖pi/pi)=rΨ(w)Φ(w).
Furthermore, from (A3), there exist two constants γ,υ∈ℝ with
(2.19)0<γ<∫01sup⁡(t1,…,tn)∈K(kr)⁡F(x,t1,…,tn)dxr,
such that
(2.20)kF(x,t1,…,tn)≤γ∑i=1n|ti|pipi+υ,∀x∈[0,1],∀(t1,…,tn)∈Rn.
Fix (u1,…,un)∈X, Then
(2.21)F(x,u1(x),…,un(x))≤1k(γ∑i=1n|ui(x)|pipi+υ)∀x∈[0,1].
So, for any fixed λ∈Λr, from (2.15) and (2.21), we have
(2.22)Φ(u)-λΨ(u)=∑i=1n‖ui′‖pipipi-λ∫01F(x,u1(x),…,un(x))dx≥∑i=1n‖ui′‖pipipi-λγk(∑i=1n1pi∫01|ui(x)|pidx)-λυk≥∑i=1n‖ui′‖pipipi-λγk(k∑i=1n‖ui′‖pipipi)-λυk=∑i=1n‖ui′‖pipipi-λγ∑i=1n‖ui′‖pipipi-λυk≥(1-γr∫01sup⁡(t1,…,tn)∈K(kr)⁡F(x,t1,…,tn)dx)∑i=1n‖ui′‖pipipi-λυk,
and thus,
(2.23)lim⁡‖(u1,…,un)‖→+∞(Φ(u1,…,un)-λΨ(u1,…,un))=+∞,
which means that the functional Φ-λΨ is coercive. So, all assumptions of Theorem 2.1 are satisfied. Hence, from Theorem 2.1 with x¯=w, taking into account that the weak solutions of the system (1.1) are exactly the solutions of the equation Φ′(u1,…,un)-λΨ′(u1,…,un)=0 and using Lemma 2.4, we have the conclusion.Now we want to present a verifiable consequence of the main result where the test functionw is specified.Put(2.24)σi=[2pi-1(x11-pi|1-∑j=1maj|pi+(1-xm)1-pi|1-∑j=1mbj|pi)]1/pifor1≤i≤n.Define(2.25)B1,n(x)={[x∑j=1maj,x]nif∑j=1maj<1,[x,x∑j=1maj]nif∑j=1maj>1,B2,n(x)={[x∑j=1mbj,x]nif∑j=1mbj<1,[x,x∑j=1mbj]nif∑j=1mbj>1,
where [·,·]n=[·,·]×⋯×[·,·], then we have the following consequence of Theorem 2.5.Corollary 2.6.
Assume that there exist2m constants  aj,bj for  1≤j≤m with  ∑j=1maj≠1 and  ∑j=1mbj≠1 and two positive constants  θ and  τ with  ∑i=1n((σiτ)pi/pi)>θ/k∏i=1npi such that (B1)
F(x,t1,…,tn)≥0 for each x∈[0,x1/2]∪[(1+xm)/2,1] and (t1,…,tn)∈B1,n(τ)∪B2,n(τ),(B2)
∑i=1n((σiτ)pi/pi)∫01sup(t1,…,tn)∈K(θ/∏i=1npi)F(x,t1,…,tn)dx<  (θ/k∏i=1npi)∫x1/2(1+xm)/2F(x,τ,…,τ)dx, where σi is given by (2.24) and K(θ/∏i=1npi)={(t1,…,tn)∣∑i=1n(|ti|pi/pi)≤c/∏i=1npi}(see (2.11));(B3)
limsup|t1|→+∞,…,|tn|→+∞(F(x,t1,…,tn)/∑i=1n(|ti|pi/pi))<  ∏i=1npi/θ×∫01sup(t1,…,tn)∈K(θ/∏i=1npi)F(x,t1,…,tn)dx uniformly with respect to x∈[0,1].then, for each λ∈]∑i=1n((σiτ)pi/pi)/∫x1/2(1+xm)/2F(x,τ,…,τ)dx,(θ/k∏i=1npi)/∫01sup(t1,…,tn)∈K(θ/∏i=1npi)F(x,t1,…,tn)dx[ the systems (1.1) admits at least three distinct classical solutions.Proof.
Setw(x)=(w1(x),…,wn(x)) such that for 1≤i≤n,
(2.26)wi(x)={τ(∑j=1maj+2(1-∑j=1maj)x1x)ifx∈[0,x12],τifx∈[x12,1+xm2],τ(2-∑j=1mbj-xm∑j=1mbj1-xm-2(1-∑j=1mbj)1-xmx)ifx∈[1+xm2,1],
and r=θ/k∏i=1npi. It is easy to see that w=(w1,…,wn)∈X, and, in particular, one has
(2.27)‖wi′‖pipi=(σiτ)pi,
for 1≤i≤n, which, employing the condition ∑i=1n((σiτ)pi/pi)>θ/k∏i=1npi, gives
(2.28)∑i=1n‖wi′‖pipipi>r.
Since for 1≤i≤n,
(2.29)τ∑j=1maj≤wi(x)≤τforeachx∈[0,x12]if∑j=1maj<1,τ≤wi(x)≤τ∑j=1majforeachx∈[0,x12]if∑j=1maj>1,τ∑j=1mbj≤wi(x)≤τforeachx∈[1+xm2,1]if∑j=1mbj<1,τ≤wi(x)≤τ∑j=1mbjforeachx∈[1+xm2,1]if∑j=1mbj>1,
the condition (B1) ensures that
(2.30)∫0x1/2F(x,w1(x),…,wn(x))dx+∫(1+xm)/21F(x,w1(x),…,wn(x))dx≥0.
Therefore, owing to assumption (B2), we have
(2.31)∫01sup⁡(t1,…,tn)∈K(kr)F(x,t1,…,tn)dx<θ(∑i=1n((σiτ)pi/pi))(k∏i=1npi)∫x1/2(1+xm)/2F(x,τ,…,τ)dx≤θk∫01F(x,w1(x),…,wn(x))dx∑i=1n∏j=1,j≠inpj‖wi′‖pipi=(r∏i=1npi)∫01F(x,w1(x),…,wn(x))dx∑i=1n∏j=1,j≠inpj‖wi′‖pipi,
where K(θ/∏i=1npi)={(t1,…,tn)∣∑i=1n(|ti|pi/pi)≤θ/∏i=1npi}. Moreover, from assumption (B3) it follows that assumption (A3) is fulfilled. Hence, taking into account that
(2.32)]∑i=1n((σiτ)pi/pi)∫x1/2(1+xm)/2F(x,τ,…,τ)dx,θ/k∏i=1npi∫01sup⁡(t1,…,tn)∈K(θ/∏i=1npi)⁡F(x,t1,…,tn)dx[⊆Λr,
using Theorem 2.5, we have the desired conclusion.Let us present an application of Corollary2.6.Example 2.7.
LetF:[0,1]×ℝ3→ℝ be the function defined as
(2.33)F(x,t1,t2,t3)={0∀ti<0,i=1,2,x2t1100e-t1fort1≥0,t2<0,t3<0,x2t2100e-t2fort1<0,t2≥0,t3<0,x2t3100e-t3fort1<0,t2<0,t3≥0.x2∑i=13ti100e-tiforti≥0,i=1,2,3,
for each (x,t1,t2,t3)∈[0,1]×ℝ3. In fact, by choosing p1=p2=p3=3 and a1=b1=x1=1/2, by a simple calculation, we obtain that k=27/8 and σ1=σ2=σ3=41/3, and so with θ=9 and τ=100, we observe that the assumptions (B1) and (B3) in Corollary 2.6 are satisfied. For (B2),
(2.34)∑i=13(σiτ)pipi∫01sup⁡(t1,t2,t3)∈K(θ/∏i=13pi)F(x,t1,t2,t3)dx=4(100)3∫01sup⁡(t1,t2,t3)∈K(1/3)F(x,t1,t2,t3)dx≤4(100)3∫01sup⁡(t1,t2,t3)∈K(1/3)x2∑i=13ti100e-tidx=4(100)3max⁡(t1,t2,t3)∈K(1/3)∑i=13ti100e-ti∫01x2dx≤43(100)3(3max⁡|t|≤1⁡t100e-t)=4(100)3e<134×34(100)100e-100=θk∏i=13pi∫x1/2(1+x1)/2F(x,τ,τ,τ)dx.
So, for every
(2.35)λ∈]44(100)326(100)100e-100,835(100)3e[,
Corollary 2.6 is applicable to the system
(2.36)-(|u1′|u1′)′=λx2(u1+)99e-u1+(100-u1+)in(0,1),-(|u2′|u2′)′=λx2(u2+)99e-u2+(100-u2+)in(0,1),-(|u3′|u3′)′=λx2(u3+)99e-u3+(100-u3+)in(0,1),ui(0)=ui(1)=12ui(12)fori=1,2,3,
where t+=max{t,0}.Here is a remarkable consequence of Corollary2.6.Corollary 2.8.
LetF:ℝn→ℝ be a  C1 function in  Rn such that  F(0,…,0)=0. Assume that there exist  2m constants  aj,bj for  1≤j≤m with  ∑j=1maj≠1 and  ∑j=1mbj≠1 and two positive constants  θ and  τ with  ∑i=1n((σiτ)pi/pi)>θ/k∏i=1npi such that (C1)
F(t1,…,tn)≥0 for each (t1,…,tn)∈B1,n(τ)∪B2,n(τ),(C2)
∑i=1n((σiτ)pi/pi)max(t1,…,tn)∈K(θ/∏i=1npi)F(t1,…,tn)<(θ(1+xm-x1)/2k∏i=1npi)F(τ,…,τ)where σi is given by (2.24),(C3)
limsup|t1|→+∞,…,|tn|→+∞(F(t1,…,tn)/∑i=1n(|ti|pi/pi))<(∏i=1npi/θ)×max(t1,…,tn)∈K(θ/∏i=1npi)F(t1,…,tn),Then, for each λ∈]∑i=1n((σiτ)pi/pi)/((1+xm-x1)/2)F(τ,…,τ),(θ/k∏i=1npi)/max(t1,…,tn)∈K(θ/∏i=1npi)F(t1,…,tn)[, the systems
(2.37)-(|ui′|pi-2ui′)′=λFui(u1,…,un)in(0,1),ui(0)=∑j=1majui(xj),ui(1)=∑j=1mbjui(xj),
for 1≤i≤n, admits at least three distinct classical solutions.Proof.
SetF(x,t1,…,tn)=F(t1,…,tn) for all x∈[0,1] and ti∈ℝ for 1≤i≤n. From the hypotheses, we see that all assumptions of Corollary 2.6 are satisfied. So, we have the conclusion by using Corollary 2.6.Example 2.9.
Letp1=p2=3,m=2,x1=1/3,x2=2/3 and ai=bi=1/3, i=1,2. Consider the system
(2.38)-(|u1′|u1′)′=λ(e-u1u111(12-u1)),in(0,1),-(|u2′|u2′)′=λ(e-u2u29(10-u2)),in(0,1),u1(0)=u1(1)=13u1(13)+13u1(23),u2(0)=u2(1)=13u2(13)+13u2(23).
Clearly, (H1) and (H2) hold. A simple calculation shows that k=125/8 and σ1=σ2=(8/3)1/3. So, by choosing θ=3 and τ=10, we observe that the assumptions (C1) and (C3) in Corollary 2.8 hold. For (C3), since F(t1,t2)=t112e-t1+t210e-t2 for every (t1,t2)∈ℝ2, we have
(2.39)max⁡(t1,t2)∈K(θ/∏i=12pi)F(t1,t2)=max⁡(t1,t2)∈K(1/3)F(t1,t2)=max⁡(t1,t2)∈K(1/3)(t112e-t1+t210e-t2)≤[max⁡|t1|≤1t112e-t1+max⁡|t2|≤1t210e-t2]=2e<1125⋅103(1012e-10+1010e-10)=θ(1+x2-x1)(∑i=12((τσi)pi/pi))(2k∏i=12pi)F(τ,τ).
Note that lim|t1|→∞,|t2|→∞(F(t1,t2)/∑i=12(|ti|pi/pi))=0. We see that for every
(2.40)λ∈]83(109e-10+107e-10),4375e[,
Corollary 2.8 is applicable to the system (2.38).Finally, we prove the theorem in the introduction.Proof of Theorem1.1.
Sincef and g are positive, then F is nonnegative in ℝ2. Fix λ>λ*:=((σ1τ)p/p+(σ2τ)q/q)/((1+x2-x1)/2)F(τ,τ) for some τ>0. Note that liminf(ξ,η)→(0,0)(F(ξ,η)/(|ξ|p/p+|η|q/q))=0, and there is {θn}n∈N⊆]0,+∞[such that limn→+∞θn=0 and
(2.41)lim⁡n→+∞max⁡(ξ,η)∈K(θn/pq)⁡F(ξ,η)θn=0.
In fact, one has limn→+∞(max(ξ,η)∈K(θn/pq)F(ξ,η)/θn)=limn→+∞(F(ξθn,ηθn)/(|ξθn|p/p+|ηθn|q/q))·(|ξθn|p/p+|ηθn|q/q)/θn=0, where F(ξθn,ηθn)=sup(ξ,η)∈K(θn/pq)F(ξ,η). Hence, there is θ¯>0 such that
(2.42)max⁡(ξ,η)∈K(θ¯/pq)⁡F(ξ,η)θ¯<min⁡{(1+x2-x1)2(q(σ1τ)p+p(σ2τ)q)kF(τ,τ);1λpqk},
and θ¯<k(q(σ1τ)p+p(σ2τ)q).
from Corollary2.8, with n=2 follows the conclusion.



---

*Source: 389530-2012-03-26.xml*