University of Mazandaran, Iran http://cjms.journals.umz.ac.ir ISSN: 1735-0611 CJMS.8(1)(2019), 43-50
Existence of multiple solutions to a two-point boundary value system via variational method
Armin Hadjian1 and Mohsen Rostamian Delavar
Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord 94531, Iran
Abstract. In this paper, we prove the existence of an open
inter-val ]λ′, λ′′[ for eachλof which a class of two-point boundary value equations depending onλadmits at least three solutions. Our main tool is a three critical points theorem of Averna and Bonanno.
Keywords: Three solutions, critical points, two-point boundary value system.
2010 Mathematics subject classification: 34B15; Secondary 58E05.
1. Introduction
Let us consider the following quasilinear elliptic system{
−ui′′+ai(x)ui=λFui(x, u1, . . ., un) in (0,1),
ui′(0) =ui′(1) = 0, (1.1)
for 1 ≤ i ≤ n, where λ is a positive parameter, F: [0,1]×Rn → R is a function such that the mapping (t1, t2, . . ., tn) → F(x, t1, t2, . . ., tn) is measurable in [0,1] for all (t1, . . ., tn) ∈ Rn and is C1 in Rn for a.e.
x∈[0,1] satisfying the condition
sup ∑n
i=1|ti|2≤ρ
|F(·, t1, . . ., tn)| ∈L1([0,1])
for every ρ >0, Fui denotes the partial derivative ofF with respect to
ui, and ai∈L∞([0,1]) with ess inf(0,1)ai>0 for 1≤i≤n.
1Corresponding author: [email protected], [email protected]
Received: 30 October 2018 Accepted: 08 December 2018
Throughout this paper, we letXbe the Cartesian product ofncopies ofW1,2([0,1]), i.e.,X= (W1,2([0,1]))n equipped with the norm
∥(u1, . . ., un)∥:= n ∑
i=1
∥ui∥,
where
∥ui∥:= (∫ 1
0
(
|ui′(x)|2+ai(x)|ui(x)|2 )
dx
)1/2
for 1≤i≤n, which is equivalent to the usual one. Put
c:= max
ui∈W1,sup2([0,1])\{0} max
x∈[0,1]|ui(x)| 2
∥ui∥2
: for 1≤i≤n
. (1.2)
Note that X is compactly embedded in (C0([0,1]))n, so c < +∞. It follows from Proposition 4.1 of [1] that
sup ui∈W1,2([0,1])\{0}
max
x∈[0,1]|ui(x)| 2
∥ui∥2
> 1
∥ai∥1
for 1≤i≤n,
where∥ai∥1:=
∫1
0 |ai(x)|dxfor 1≤i≤n,and so 1
∥ai∥1 ≤c for 1≤i≤n.
By a(weak) solution of system (1.1), we mean anyu= (u1, . . ., un)∈
X such that ∫ 1
0
n ∑
i=1
ui′(x)vi′(x)dx+ ∫ 1
0
n ∑
i=1
ai(x)ui(x)vi(x)dx
−λ
∫ 1
0
n ∑
i=1
Fui(x, u1(x),. . ., un(x))vi(x)dx= 0
for all v= (v1, . . ., vn)∈X.
We shall establish the existence of a definite interval, in whichλlies, system (1.1) admits at least three weak solutions in X, by means of a recent abstract critical points result of Averna and Bonanno [2] which is actually a refinement of a general principle of Ricceri [8]. The existence and multiplicity of solutions for two-point boundary value problems have been widely investigated (see, for instance, [3, 4, 5, 6, 7] and references therein). For other basic notations and definitions we refer to [9].
2. Main result
First we here recall for the reader’s convenience the three critical points theorem of [2] which is our main tool to prove the results. Here,
Theorem 2.1 (Theorem B of [2]). Let Y be a real reflexive Banach space; Φ : Y → R a continuously Gˆateaux differentiable and sequen-tially weakly lower semicontinuous functional whose Gˆateaux derivative admits a continuous inverse onY∗;Ψ :Y →Ra continuously Gˆateaux differentiable functional whose Gˆateaux derivative is compact. Assume that
(i) lim∥u∥→+∞(Φ(u) +λΨ(u)) = +∞ for allλ∈[0,+∞[;
(ii) there is r∈R such that:
inf Y Φ< r, and
φ1(r)< φ2(r),
where
φ1(r) := inf
u∈Φ−1(]−∞,r[)
Ψ(u)−infΦ−1(]−∞,r[)wΨ r−Φ(u) ,
φ2(r) := inf
u∈Φ−1(]−∞,r[) sup
v∈Φ−1([r,+∞[)
Ψ(u)−Ψ(v) Φ(v)−Φ(u),
and Φ−1(]− ∞, r[)w is the closure ofΦ−1(]− ∞, r[) in the weak
topology.
Then, for eachλ∈]φ1
2(r),
1
φ1(r)[ the functional Φ +λΨ has at least three
critical points in Y.
For allγ >0 we denote byK(γ) the set {
(t1, . . ., tn)∈Rn : n ∑
i=1
|ti|2 ≤γ }
. (2.1)
We formulate our main result as follows.
Theorem 2.2. Assume that there exist two positive constantsγ andδ
withδ2 > γ
2n−1 such that
(j) ∫1
0 sup
(t1,...,tn)∈K( γ 2n−1)
F(x,t1,...,tn)dx
γ <
∫1
0 F(x,δ,...,δ)dx
2ncδ2∑n
i=1∥ai∥1,wherecandK(
γ
2n−1)
are given by (1.2)and (2.1); (jj) lim sup
|t1|→+∞,...,|tn|→+∞
F(x,t1,...,tn) ∑n
i=1|ti| 2
2
≤ 0 uniformly with respect to x ∈
[0,1];
Then, setting
λ′:=
δ2
2
∑n
i=1∥ai∥1
∫1
0 F(x, δ, . . ., δ) −
∫1
0 sup
(t1,...,tn)∈K( γ
2n−1)
F(x, t1, . . ., tn)dx
and
λ′′:= γ
2nc∫1
0 sup
(t1,...,tn)∈K(2nγ−1)
F(x, t1, . . ., tn)dx
,
for eachλ∈]λ′, λ′′[system (1.1)admits at least three weak solutions in
X.
Proof. For each u= (u1, . . . , un)∈X, put
Φ (u) := n ∑
i=1
∥ui∥2
2 and Ψ(u) :=− ∫ 1
0
F(x, u1(x), . . . , un(x))dx.
It is well known that Φ and Ψ are well defined and continuously Gˆateaux differentiable functionals with
Φ′(u)(v) = ∫ 1
0
n ∑
i=1
ui′(x)vi′(x)dx+ ∫ 1
0
n ∑
i=1
ai(x)ui(x)vi(x)dx
and
Ψ′(u)(v) =− ∫ 1
0
n ∑
i=1
Fui(x, u1(x),. . ., un(x))vi(x)dx
for everyu= (u1, . . ., un),v = (v1, . . ., vn)∈X, as well as Ψ′:X→X∗ is continuous and compact operator, and Φ′ :X→X∗ admits a continuous inverse onX∗.Furthermore, by Proposition 25.20 of [9], Φ is sequentially weakly lower semicontinuous. Thanks to the assumption (jj), for each
λ >0 one has
lim
∥u∥→+∞(Φ(u) +λΨ(u)) = +∞.
Putr:= 2γnc.From the hypothesis (j), we get ∫1
0 sup
(t1,...,tn)∈K( γ
2n−1)
F(x, t1, . . ., tn)dx
γ <
∫1
0 F(x, δ, . . ., δ)dx
2n−1cδ2∑n
i=1∥ai∥1
−
∫1
0 sup
(t1,...,tn)∈K( γ
2n−1)
F(x, t1, . . ., tn)dx
Thus, sinceδ2> 2nγ−1,and c∥ai∥1 ≥1 for 1≤i≤n, we have ∫1
0 sup
(t1,...,tn)∈K( γ
2n−1)
F(x, t1, . . ., tn)dx
γ
<
∫1
0 F(x, δ, . . ., δ)dx−
∫1
0 sup
(t1,...,tn)∈K(2nγ−1)
F(x, t1, . . ., tn)dx
2n−1cδ2∑n
i=1∥ai∥1
,
from which, multiplying by 2nc, we obtain 2nc∫01 sup
(t1,...,tn)∈K( γ
2n−1)
F(x, t1, . . ., tn)dx
γ (2.2)
<
∫1
0 F(x, δ, . . ., δ)dx−
∫1
0 sup
(t1,...,tn)∈K(2nγ−1)
F(x, t1, . . ., tn)dx
δ2
2
∑n
i=1∥ai∥1
.
We claim that
φ1(r)≤
2nc∫1
0 sup
(t1,...,tn)∈K( γ
2n−1)
F(x, t1, . . ., tn)dx
γ (2.3)
and
φ2(r)≥
∫1
0 F(x, δ, . . ., δ)dx −
∫1
0 sup
(t1,...,tn)∈K(2nγ−1)
F(x, t1, . . ., tn)dx
δ2
2
∑n i=1∥ai∥1
, (2.4)
from which (ii) of Theorem 2.1 follows.
In fact, taking into account that the function identically 0 obviously belongs to Φ−1(]− ∞, r[),and that Ψ(0) = 0,we get
φ1(r)≤
supΦ−1(]−∞,r[)w
∫1
0 F(x, u1(x),. . ., un(x))dx
r ,
and, since Φ−1(]− ∞, r[)w = Φ−1(]− ∞, r]),we have
supΦ−1(]−∞,r[)w∫1
0 F(x, u1(x),. . ., un(x))dx
r
=
sup
Φ−1(]−∞,r])
∫1
0 F(x, u1(x),. . ., un(x))dx
r .
Since for eachui∈W1,2([0,1]) sup x∈[0,1]
for 1≤i≤n (see (1.2)), we have that
sup x∈[0,1]
n ∑
i=1
|ui(x)|2
2 ≤c
n ∑
i=1
∥ui∥2
2 =cΦ(u)
for every u = (u1, . . ., un) ∈ X. Thus, taking into account that ∑n
i=1|ui(x)|2≤
γ
2n−1,for everyu= (u1, . . ., un)∈Xsuch that Φ(u)≤r and for eachx∈[0,1],we obtain
sup
Φ−1(]−∞,r])
∫1
0 F(x, u1(x),. . ., un(x))dx
r
≤
∫1
0 sup
(t1,...,tn)∈K( γ
2n−1)
F(x, t1, . . ., tn)dx
r .
So, (2.3) follows at once by the definition ofr.
Moreover, for eachv= (v1, . . ., vn)∈X such that Φ(v)≥r, we have
φ2(r)≥
inf u∈Φ−1(]−∞,r[)
∫1
0 F(x, v1(x), . . ., vn(x))dx −
∫1
0 F(x, u1(x),. . ., un(x))dx
Φ(v)−Φ(u) ,
thus, from ∑ni=1|ui(x)|2 ≤ 2nγ−1, for every u = (u1, . . ., un) ∈ X such that Φ(u)< rand for each x∈[0,1],we obtain
inf
u∈Φ−1(]−∞,r[) ∫1
0 F(x, v1(x), . . ., vn(x))dx −
∫1
0 F(x, u1(x),. . ., un(x))dx Φ(v)−Φ(u)
≥ inf u∈Φ−1(]−∞,r[)
∫1
0 F(x, v1(x), . . ., vn(x))dx − ∫01 sup
(t1,...,tn)∈K(2nγ−1)
F(x, t1, . . ., tn)dx
Φ(v)−Φ(u) ,
from which, being 0<Φ(v)−Φ(u)≤Φ(v) for everyu∈Φ−1(]− ∞, r[),
and under further condition ∫ 1
0
F(x, v1(x),. . ., vn(x))dx≥ ∫ 1
0
sup (t1,...,tn)∈K(2nγ−1)
F(x, t1, . . ., tn)dx, (2.5)
we can write
inf
u∈Φ−1(]−∞,r[) ∫1
0 F(x, v1(x),. . ., vn(x))dx−
∫1
0 sup
(t1,...,tn)∈K(2nγ−1)
F(x, t1, . . ., tn)dx
Φ(v)−Φ(u)
≥
∫1
0 F(x, v1(x),. . ., vn(x))dx−
∫1
0 sup
(t1,...,tn)∈K(2nγ−1)
F(x, t1, . . ., tn)dx ∑n
i=1∥
vi∥2 2
.
If we put v(x) := (δ, . . ., δ),for each x ∈[0,1], we have ∥vi∥=∥ai∥11/2δ
Now since δ2 > 2nγ−1, bearing in mind that ∥a1
i∥1 ≤c for 1≤ i≤ n,
we get Φ(v) =δ22∑ni=1∥ai∥1 > r. Moreover, with this choice of v, (2.2)
ensures (2.5), thus (2.4) is also proved.
Taking into account that the weak solutions of system (1.1) are exactly the solutions of the equation Φ′(u) +λΨ′(u) = 0,we have the conclusion by using of Theorem 2.1. Namely, by observing that
1
φ2(r) ≤
δ2
2
∑n
i=1∥ai∥1
∫1
0 F(x, δ, . . ., δ)dx −
∫1
0 sup(t1,...,tn)∈K( γ
2n−1)
F(x, t1, . . ., tn)dx
and
1
φ1(r) ≥
γ
2nc∫1
0 sup
(t1,...,tn)∈K( γ
2n−1)
F(x, t1, . . ., tn)dx
,
for eachλ∈]λ′, λ′′[ system (1.1) admits at least three weak solutions in
X.
It is of interest to list some special cases of Theorem 2.2.
Theorem 2.3. LetF :Rn→Rbe aC1-function and assume that there exist two positive constantsγ andδ with δ2> γ
2n−1 such that
(j′)
max
(t1,...,tn)∈K( γ 2n−1)
F(t1,...,tn)
γ <
F(δ,...,δ) 2ncδ2∑n
i=1∥ai∥1, wherecandK(
γ
2n−1)
are given by (1.2)and (2.1); (jj′) lim sup
|t1|→+∞,...,|tn|→+∞
F(t1,...,tn) ∑n
i=1 |ti|2
2
≤0;
(jjj′) F(0, . . . ,0) = 0.
Then, setting
λ′ :=
δ2
2
∑n
i=1∥ai∥1
F(δ, . . ., δ) − max
(t1,...,tn)∈K( γ
2n−1)
F(t1, . . ., tn)
and
λ′′:= γ
(2nc) max
(t1,...,tn)∈K(2nγ−1)
F(t1, . . ., tn)
,
for each λ∈]λ′, λ′′[ the system {
−ui′′+ai(x)ui =λFui(u1, . . ., un) in (0,1),
ui′(0) =ui′(1) = 0,
Corollary 2.4. Let f :R → R be a continuous function. Put F(t) = ∫t
0f(ξ)dξ for each t∈ R and assume that there exist two positive
con-stantsγ andδ with δ2> γ such that
(j′′)
max t∈[−√γ,√γ]F(t)
γ < F(δ) 2cδ2∥a∥
1, where c:= sup
u∈W1,2([0,1])\{0}
(∥u∥∞ ∥u∥
)2
;
(jj′′) lim sup |t|→+∞
F(t)
|t|2 ≤0.
Then, setting
λ′ := δ
2∥a∥ 1
2(F(δ)− max
t∈[−√γ,√γ]F(t))
and
λ′′ := γ
(2c) max
t∈[−√γ,√γ]F(t)
,
for each λ∈]λ′, λ′′[ the problem {
−u′′+a(x)u=λf(u) in (0,1), u′(0) =u′(1) = 0
admits at least three classical solutions in W1,2([0,1]).
Acknowledgment. This research was in part supported by a grant from University of Bojnord (No. 97/367/4878).
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