Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 56, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS FOR DIRICHLET QUASILINEAR
SYSTEMS INCLUDING A NONLINEAR FUNCTION OF THE
DERIVATIVE
SHAPOUR HEIDARKHANI, MASSIMILIANO FERRARA, GHASEM A. AFROUZI,
GIUSEPPE CARISTI, SHAHIN MORADI
Abstract. In this article we establish the existence of at least one non-trivial
classical solution for Dirichlet quasilinear systems with a nonlinear dependence
on the derivative. We use variational methods for smooth functionals defined
on reflexive Banach spaces, and assumptions on the asymptotic behaviour of
the nonlinear data.
1. Introduction
In this article we consider the quasilinear system
−(pi − 1)|u0i (x)|pi −2 u00i (x) = λFui (x, u1 , . . . , un )hi (x, u0i ),
ui (a) = ui (b) = 0
x ∈ (a, b),
(1.1)
for 1 ≤ i ≤ n where pi > 1 for 1 ≤ i ≤ n, λ > 0, a, b ∈ R with a < b, F : [a, b] ×
Rn → R is measurable with respect to x, for every (t1 , . . . , tn ) ∈ Rn , continuously
differentiable in (t1 , . . . , tn ), for almost every x ∈ [a, b], and Fti (x, 0, . . . , 0) = 0 for
all x ∈ [a, b] and for 1 ≤ i ≤ n, hi : [a, b]×R → [0, +∞[ is a bounded and continuous
function with mi := inf (x,t)∈[a,b]×R hi (x, t) > 0 for 1 ≤ i ≤ n. Here, Fti denotes the
partial derivative of F with respect to ti .
Owing to the importance of second-order differential equations with nonlinear
derivative dependence in physics, in the last decade or so, many authors applied
the variational method to study the existence of solutions of the problems of the
form of (1.1) or its variations, see, for example, [2, 3, 5, 8, 9, 10, 11]. We note that
the main tools in these cited papers are critical points theorems due to Ricceri and
their variants.
Our goal is to establish some new criteria for system (1.1) to have at least one
non-trivial classical solution by applying the following critical points theorem due
to Ricceri [15, Theorem 2.1].
2010 Mathematics Subject Classification. 34B15, 58E05.
Key words and phrases. Classical solution; Dirichlet quasilinear system; critical point theory;
variational methods.
c
2016
Texas State University.
Submitted November 17, 2015. Published February 25, 2016.
1
2 S. HEIDARKHANI, M. FERRARA, G. A. AFROUZI, G. CARISTI, S. MORADI
EJDE-2016/56
Theorem 1.1. Let X be a reflexive real Banach space, let Φ, Ψ : X → R be two
Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous and coercive in X and Ψ is sequentially weakly upper
semicontinuous in X. Let Iλ be the functional defined as Iλ := Φ − λΨ, λ ∈ R, and
for every r > inf X Φ, let ϕ be the function defined as
ϕ(r) :=
supv∈Φ−1 (−∞,r) Ψ(v) − Ψ(u)
.
r − Φ(u)
u∈Φ−1 (−∞,r)
inf
1
), the restriction of the functional
Then, for every r > inf X Φ and every λ ∈ (0, ϕ(r)
−1
Iλ to Φ (−∞, r) admits a global minimum, which is a critical point (precisely a
local minimum) of Iλ in X.
We refer the interested reader to the papers [1, 6, 7, 12, 13, 14] in which Theorem
1.1 has been used for the existence of at least one nontrivial solution for boundary
value problem.
2. Preliminaries
Let X be the Cartesian product of n Sobolev spaces W01,p1 ([a, b]), . . . , and
i.e., X = W01,p1 ([a, b]) × · · · × W01,pn ([a, b]), equipped with the norm
W01,pn ([a, b]),
k(u1 , . . . , un )k =
n
X
ku0i kpi
i=1
where
ku0i kpi =
Z
b
1/pi
|u0i (x)|pi dx
,
i = 1, . . . , n.
a
Since pi > 1 for i = 1, . . . , n, X is compactly embedded in (C([a, b]))n .
By a classical solution of (1.1), we mean a function u = (u1 , . . . , un ) such that,
for i = 1, . . . , n, ui ∈ C 1 [a, b], u0i ∈ AC[a, b], and ui (x) satisfies (1.1) a.e. on [a, b].
We say that a function u = (u1 , . . . , un ) ∈ X is a weak solution of the system (1.1)
if
n Z b Z u0i (x)
X
(pi − 1)|τ |pi −2 0
dτ vi (x)dx
hi (x, τ )
0
i=1 a
Z bX
n
−λ
Fui (x, u1 (x), . . . , un (x))vi (x)dx = 0
a i=1
for every v = (v1 , . . . , vn ) ∈ X. Using standard methods, we see that a weak
solution of system (1.1) is indeed a classical solution (see [8, Lemma 2.2]).
Let
p := min{pi : 1 ≤ i ≤ n},
mi :=
Mi :=
inf
(x,t)∈[a,b]×R
sup
p := max{pi : 1 ≤ i ≤ n},
hi (x, t) > 0,
hi (x, t),
for 1 ≤ i ≤ n,
textf or1 ≤ i ≤ n,
(x,t)∈[a,b]×R
M := max{Mi : 1 ≤ i ≤ n},
M := min{mi : 1 ≤ i ≤ n}.
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EXISTENCE OF SOLUTIONS
3
Then, M ≥ Mi ≥ mi ≥ M > 0 for each i = 1, . . . , n. Put
Z tZ τ
(pi − 1)|δ|pi −2 Hi (x, t) =
dδ dτ
hi (x, δ)
0
0
for all (x, t) ∈ [a, b] × R, 1 ≤ i ≤ n.
3. Main results
We formulate our min result as follows. Put
(
p if b − a ≥ 1,
∗
p =
p if 0 < b − a < 1.
Theorem 3.1. Assume that
sup R b
r>0
a
r
max
>1
∗
(b−a)p −1 pM r
)
(t1 ,...,tn )∈Q(
p
2
(3.1)
F (x, t1 , . . . , tn )dx
where
Q(
∗
∗
n
n
X
(b − a)p −1 pM r
|ti |pi
(b − a)p −1 pM r o
n
)
=
(t
,
.
.
.
,
t
)
∈
R
:
≤
1
n
2p
pi
2p
i=1
and there are a non-empty open set D ⊆ (a, b) and B ⊂ D of positive Lebesgue
measure such that
ess inf x∈B F (x, ξ1 , . . . , ξn )
Pn
lim sup
= +∞,
p
(ξ1 ,...,ξn )→(0+ ,...,0+ )
i=1 |ξi |
ess inf x∈D F (x, ξ1 , . . . , ξn )
Pn
> −∞.
p
i=1 |ξi |
lim inf
(ξ1 ,...,ξn )→(0+ ,...,0+ )
Then, for each
λ ∈ Λ = 0, sup R b
r>0
a
r
max
(t1 ,...,tn )∈Q(
∗
(b−a)p −1 pM r
)
p
2
,
F (x, t1 , . . . , tn )dx
system (1.1) admits at least one non-trivial classical solution uλ = (u1λ , . . . , unλ )
in X. Moreover, we have
lim+ kuλ k = 0
λ→0
and the real function
Z
n Z b
X
0
λ→
Hi (x, uiλ (x))dx − λ
i=1
a
b
F (x, u1λ (x), . . . , unλ (x))dx
a
is negative and strictly decreasing in the open interval Λ.
Proof. Our aim is to apply Theorem 1.1 to system (1.1). Let the functionals Φ, Ψ
be defined by
n Z b
X
Φ(u) =
Hi (x, u0i (x))dx,
(3.2)
Z
i=1
b
Ψ(u) =
a
F (x, u1 (x), . . . , un (x))dx
a
for u = (u1 , . . . , un ) ∈ X, and put
Iλ (u) = Φ(u) − λΨ(u)
(3.3)
4 S. HEIDARKHANI, M. FERRARA, G. A. AFROUZI, G. CARISTI, S. MORADI
EJDE-2016/56
for u ∈ X. Let us prove that the functionals Φ and Ψ satisfy the required conditions in Theorem 1.1. It is well known that Ψ is a differentiable functional whose
differential at the point u ∈ X is
Z bX
n
Ψ0 (u)(v) =
Fui (x, u1 (x), . . . , un (x))vi (x)dx
a i=1
for every v = (v1 , . . . , vn ) ∈ X, and Ψ is sequentially weakly upper semicontinuous.
Moreover, according to the definition Φ, we see that Φ is continuous. Since 0 <
M ≤ hi (x, t) ≤ M for each (x, t) ∈ [a, b] × R and i = 1, . . . , n, from (3.2) we see
that
n
n
1 X ku0i kppii
1 X ku0i kppii
≤ Φ(u) ≤
for all u ∈ X.
(3.4)
M i=1 pi
M i=1 pi
From (3.4), it follows limkuk→+∞ Φ(u) = +∞, namely Φ is coercive. Moreover, Φ
is continuously differentiable whose differential at the point u ∈ X is
n Z b Z u0i (x)
X
(pi − 1)|τ |pi −2 0
0
dτ vi (x)dx
Φ (u)(v) =
hi (x, τ )
0
i=1 a
for every v ∈ X. Furthermore, Φ is sequentially weakly lower semicontinuous (see
[17, Proposition25.20]). Since for 1 ≤ i ≤ n,
(b − a)
2
max |ui (x)| ≤
x∈[a,b]
pi −1
pi
ku0i kpi
for each ui ∈ W01,pi ([a, b]) (see [16]), we have
n
X
|ui (x)|pi
max
x∈[a,b]
pi
i=1
≤
(b − a)p
2p
∗
−1
n
X
ku0i kppi
i
i=1
pi
for each u ∈ X. This, for each r > 0, along with (3.4), ensures that
Φ−1 (−∞, r)
= {u ∈ X; Φ(u) < r}
∗
n
n
o
X
|ui (x)|pi
(b − a)p −1 pM r
⊆ u ∈ X : max
≤
for each x ∈ [a, b] .
p
pi
2
i=1
(3.5)
So,
Z
max
−1
u∈Φ
Ψ(u) ≤
(−∞,r)
b
max
p∗ −1 pM r
)
p
2
F (x, t1 , . . . , tn )dx.
(3.6)
a (t1 ,...,tn )∈Q( (b−a)
From the definition of ϕ(r), since 0 ∈ Φ−1 (−∞, r) and Φ(0) = Ψ(0) = 0, one has
(supv∈Φ−1 (−∞,r) Ψ(v)) − Ψ(u)
r − Φ(u)
u∈Φ−1 (−∞,r)
supv∈Φ−1 (−∞,r) Ψ(v)
≤
r
Rb
∗
max
(b−a)p −1 pM r F (x, t1 , . . . , tn )dx
a
(t1 ,...,tn )∈Q(
)
p
2
≤
.
r
ϕ(r) =
inf
(3.7)
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EXISTENCE OF SOLUTIONS
5
Hence, putting
λ∗ = sup R b
r>0
a
r
max
,
∗
(b−a)p −1 pM r
(t1 ,...,tn )∈Q(
)
p
2
F (x, t1 , . . . , tn )dx
1
Theorem 1.1 ensures that for every λ ∈ (0, λ∗ ) ⊆ (0, ϕ(r)
), the functional Iλ admits
−1
at least one critical point (local minima) uλ ∈ Φ (−∞, r). Now for every fixed
λ ∈ (0, λ∗ ) we show that uλ = (u1λ , . . . , unλ ) 6= 0 and the map
(0, λ∗ ) 3 λ 7→ Iλ (uλ ),
is negative. To this end, let us verify that
lim sup
kuk→0+
Ψ(u)
= +∞.
Φ(u)
(3.8)
Owing to our assumptions at zero, we can fix a sequence {ξ k = (ζ k , . . . , ζ k )} ⊂
(R+ )n converging to zero and two constants σ, κ (with σ > 0) such that
ess inf x∈B F (x, ξ k )
Pn
= +∞,
k p
k→+∞
i=1 |ζ |
n
X
|ξi |p
ess inf x∈D F (x, ξ) ≥ κ
lim
i=1
where ξ = (ξ1 , . . . , ξn ), ξi ∈ [0, σ], 1 ≤ i ≤ n. Now, fix a set C ⊂ B of positive
measure and a function v = (v1 , . . . , vn ) ∈ X such that:
(i) vi (x) ∈ [0, 1], for 1 ≤ i ≤ n and for every x ∈ [a, b],
(ii) v(x) = (1, . . . , 1) for every x ∈ C,
(iii) v(x) = (0, . . . , 0) for every x ∈ (a, b) \ D.
Hence, fix M > 0 and consider a real positive number η with
R
Pn
nη meas(C) + κ D\C i=1 |vi (x)|p dx
M<
.
Pn
1
0 p
i=1 kvi kp
pM
Then, there is k0 ∈ N such that ζ k < σ and
ess inf x∈B F (x, ξ k ) ≥ η
n
X
|ζ k |p
i=1
for every k > k0 . Now, for every k > k0 , recalling the properties of the function v
(that is 0 ≤ ζ k vi (x) < σ, 1 ≤ i ≤ n for k sufficiently large), one has
R
R
F (x, ζ k , . . . , ζ k )dx + D\C F (x, ζ k v1 (x), . . . , ζ k vn (x))dx
Ψ(ξ k v)
C
=
Φ(ξ k v)
Φ(ξ k v)
R
Pn
nη meas(C) + κ D\C i=1 |vi (x)|p dx
>
>M
Pn
1
0 p
i=1 kvi kp
pM
where ξ k v = (ζ k v1 , . . . , ζ k vn ). Since M could be consider arbitrarily large, it follows
that
Ψ(ξ k v)
= +∞,
lim
k→∞ Φ(ξ k v)
6 S. HEIDARKHANI, M. FERRARA, G. A. AFROUZI, G. CARISTI, S. MORADI
EJDE-2016/56
from which (3.8) follows. Hence, there exists a sequence {wk = (w1k , . . . , wnk )} ⊂
X strongly converging to zero, wk ∈ Φ−1 (−∞, r) and
Iλ (wk ) = Φ(wk ) − λΨ(wk ) < 0.
Since uλ = (u1λ , . . . , unλ ) is a global minimum of the restriction of Iλ to the set
Φ−1 (−∞, r), we deduce that
Iλ (uλ ) < 0,
(3.9)
so that uλ is not trivial. From (3.9) we easily see that the map
(0, λ∗ ) 3 λ 7→ Iλ (uλ )
(3.10)
is negative.
Now we prove that limλ→0+ kuλ k = 0. Since Φ is coercive and for λ ∈ (0, λ∗ )
the solution uλ ∈ Φ−1 (−∞, r), one has that there exists a positive constant L such
that kuλ k ≤ L for every λ ∈ (0, λ∗ ). Therefore, there exists a positive constant M
such that
n
Z bX
Fuλi (x, u1λ (x), . . . , unλ (x))uλ (x)dx ≤ Mkuλ k ≤ ML
(3.11)
a i=1
for every λ ∈ (0, λ∗ ). Since uλ is a critical point of Iλ , we have Iλ0 (uλ )(v) = 0 for
any v ∈ X and every λ ∈ (0, λ∗ ). In particular Iλ0 (uλ )(uλ ) = 0; that is,
Z bX
n
Φ0 (uλ )(uλ ) = λ
Fuλi (x, u1λ (x), . . . , unλ (x))uλ (x)dx
(3.12)
a i=1
for every λ ∈ (0, λ∗ ). Then, since
0≤
n
1 X 0 pi
kui kpi ≤ Φ0 (uλ )(uλ )
M i=1
by (3.12) it follows that
Z bX
n
n
1 X 0 pi
0≤
kui kpi ≤ λ
Fuλi (x, u1λ (x), . . . , unλ (x))uλ (x)dx
M i=1
a i=1
for any λ ∈ (0, λ∗ ). Letting λ → 0+ , by (3.11) we have limλ→0+ kuλ k = 0. Further,
taking X ,→ (C([a, b]))n into account, one has
lim kuλ k∞ = 0.
λ→0+
(3.13)
Finally, we show that the map λ 7→ Iλ (uλ ) is strictly decreasing in (0, λ∗ ). For our
aim we see that for any u ∈ X,
Φ(u)
Iλ (u) = λ
− Ψ(u) .
(3.14)
λ
Now, let us fix 0 < λ1 < λ2 < λ∗ and let ūλ1 = (u1λ1 , . . . , unλ1 ) and ūλ2 =
(u1λ2 , . . . , unλ2 ) be the global minimum of the functional Iλi restricted to Φ(−∞, r)
for i = 1, 2. Also, let
Φ(ū )
Φ(v)
λi
mλi =
− Ψ(ūλi ) =
inf
− Ψ(v)
λi
λi
v∈Φ−1 (−∞,r)
for i = 1, 2. Clearly, (3.10) and (3.14), since λ > 0, imply that
mλi < 0,
for i = 1 2.
(3.15)
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EXISTENCE OF SOLUTIONS
7
Moreover, since 0 < λ1 < λ2 , we have
mλ2 ≤ mλ1 .
(3.16)
Hence, from (3.14)-(3.16) and again since 0 < λ1 < λ2 , we obtain
Iλ2 (ūλ2 ) = λ2 mλ2 ≤ λ2 mλ1 < λ1 mλ1 = Iλ1 (ūλ1 ),
which means the map λ 7→ Iλ (uλ ) is strictly decreasing in λ ∈ (0, λ∗ ). Since λ < λ∗
is arbitrary, we observe λ 7→ Iλ (uλ ) is strictly decreasing in (0, λ∗ ). The proof is
complete.
Remark 3.2. Here employing Ricceri’s variational principle we are looking for
the existence of critical points of the functional Iλ naturally associated to system
(1.1). We emphasize that by direct minimization, we can not argue, in general for
finding the critical points of Iλ . Because, in general, Iλ can be unbounded from
the following in X. Indeed, for example, in the case when
n
X
F (x, t1 , . . . , tn ) = ( (|ti | + |ti |qi ))
i=1
for all (x, t1 , . . . , tn ) ∈ [a, b] × Rn with qi ∈ (p, +∞) for 1 ≤ i ≤ n, for any fixed
u ∈ X\{0} and ι ∈ R, we obtain
Z b
Iλ (ιu) = Φ(ιu) − λ
F (x, ιu1 (x), . . . , ιun (x))dx
a
n
n
n
X
X
ι p X 0 pi
kui kpi − λι
kui kL1 − λ
ιqi kui kqLiqi → −∞
≤
pM i=1
i=1
i=1
as ι → +∞.
Remark 3.3. We want to point out that the energy functional Iλ associated
with system (1.1) P
is not coercive. Indeed, fix u ∈ X\{0} and ι ∈ R, then, for
n
F (x, t1 , . . . , tn ) = i=1 |ti |si for all (x, t1 , . . . , tn ) ∈ [a, b] × Rn with si ∈ (p, +∞).
We have
Z b
Iλ (ιu) = Φ(ιu) − λ
F (x, ιu1 (x), . . . , ιun (x))dx
a
n
n
X
ιp X
ιsi kui ksLisi → −∞
≤
kui kppii − λ
pM i=1
i=1
as ι → +∞.
Remark 3.4. If in Theorem 3.1, Fti (x, t1 , . . . , tn ) ≥ 0 for a.e. x ∈ [a, b], 1 ≤ i ≤ n,
the condition (3.1) becomes to the more simple and significative form
r
q
q
sup R
> 1.
(3.17)
∗ −1 2
∗
p
p (b−a)
p (b−a)p −1 p2 M r
p Mr
r>0 b
F
(x,
,
.
.
.
,
)dx
p
p
2
2
a
Moreover, if
lim sup R
r→+∞
r
b
a
q
F (x,
p
(b−a)p∗ −1 p2 M r
,...,
2p
then, (3.17) automatically holds.
q
p
(b−a)p∗ −1 p2 M r
)dx
2p
> 1,
8 S. HEIDARKHANI, M. FERRARA, G. A. AFROUZI, G. CARISTI, S. MORADI
EJDE-2016/56
Remark 3.5. For fixed r̄ > 0 let
r̄
Rb
a
max
> 1.
∗
(b−a)p −1 pM r̄
(t1 ,...,tn )∈Q(
)
p
2
F (x, t1 , . . . , tn )dx
Then the result of Theorem 3.1 holds. In fact, it ensures the existence of least one
non-trivial classical solution uλ = (u1λ , . . . , unλ ) in X such that
max
n
X
|uiλ (x)|pi
i=1
pi
∗
≤
(b − a)p −1 pM r̄
2p
for each x ∈ [a, b].
Remark 3.6. We observe that Theorem 3.1 is a bifurcation result in the sense
that the pair (0, 0) belongs to the closure of the set {(uλ , λ) ∈ X × (0, +∞) : uλ =
(u1λ , . . . , unλ ) is a non-trivial classical solution of (1.1) } in X × R. Indeed, by
Theorem 3.1 we have that
kuλ k → 0
as λ → 0.
Hence, there exist two sequences {uj = (u1j . . . , unj )} in X and {(λ1 , . . . , λn )} in
(R+ )n (here uij = uλi ) such that for 1 ≤ i ≤ n
λi → 0+
and kuij k → 0,
as j → +∞. Moreover, we want to emphasis that because the map (0, λ∗ ) 3 λ 7→
Iλ (uλ ) is strictly decreasing, for every λ1 , λ2 ∈ (0, λ? ), with λ1 6= λ2 , the solutions
ūλ1 and ūλ2 ensured by Theorem 3.1 are different.
We present the following example in which the hypotheses of Theorem 3.1 are
satisfied.
Example 3.7. Consider the system
−u001 (x) = λFu1 (u1 , u2 )h1 (u01 ),
x ∈ (0, 1),
−u002 (x)
x ∈ (0, 1),
=
λFu2 (u1 , u2 )h2 (u02 ),
u1 (0) = u1 (1) = 0,
(3.18)
u2 (0) = u2 (1) = 0
where F (t1 , t2 ) = (t21 + t22 ) sin2 (t1 ) + cos(t2 ) for

if t1

2,
2
√
h1 (t1 ) = 1+ t1 , if t1


1,
if t1
t1 , t2 ∈ R,
∈ (−∞, 0),
∈ [0, 1],
∈ (1, +∞)
4
and h2 (t2 ) = 3+sint
for t2 ∈ R. We observe that F , h1 and h2 are continuous, h1
2
and h2 are bounded with m1 = inf h1 = 1, m2 = inf h2 = 1, M1 = sup h1 = 2 and
M2 = sup h2 = 2. By the definitions of h1 and h2 we have
1
2
if t1 ∈ (−∞, 0),

 4 t1 ,
5
1
2
2
2
H1 (t1 ) = 4 t1 + 15 t1 ,
if t1 ∈ [0, 1],

1 2 1
2
t
−
t
+
,
if
t1 ∈ (1, +∞)
2 1
4 1
15
and H2 (t2 ) = 38 t22 − sin4t2 for t2 ∈ R. It is easy to see that all assumptions of
Theorem 3.1 are satisfied. By Theorem 3.1 for each λ ∈ (0, 14 , system (3.18) admits
EJDE-2016/56
EXISTENCE OF SOLUTIONS
9
at least one nontrivial classical solution uλ = (u1λ , u2λ ) in X. Moreover, we have
limλ→0+ kuλ k = 0 and the real function
Z 1
H1 (u01λ (x)) + H2 (u02λ (x)) dx
λ 7→
0
Z
1
−λ
(u21λ (x) + u22λ (x)) sin2 (u1λ (x)) + cos(u2λ (x)) dx
0
is negative and strictly decreasing in (0, 41 ).
As an application of Theorem 3.1, we consider the problem
−(p − 1)|u0 (x)|p−2 u00 (x) = λf (x, u)h(u0 ),
x ∈ (a, b),
u(a) = u(b) = 0
(3.19)
where p > 1, λ > 0, f : [a, b] × R → R is an L1 -Carathéodory function such that
f (x, 0) = 0 for a.e. x ∈ [a, b], and h : R → [0, +∞) is a bounded and continuous
function with m = inf t∈R h(t) > 0.
Let M = supt∈R h(t),
Z t
F (x, t) =
f (x, ξ)dξ for every (x, t) ∈ [a, b] × R,
0
Z tZ τ
(p − 1)|δ|p−2 H(t) =
dδ dτ
h(δ)
0
0
for all t ∈ R. For γ > 0, define
W (γ) = {t ∈ R : |t|p ≤ pγ}.
We conclude this section by giving the following consequence of Theorem 3.1.
Theorem 3.8. Assume that
sup R b
r>0
a
r
>1
maxt∈W ( (b−a)p−1 pM r ) F (x, t)dx
2p
and there are a non-empty open sets D ⊆ (0, T ) and B ⊂ D of positive Lebesgue
measure such that
ess inf x∈B F (x, ξ)
lim sup
= +∞,
ξp
ξ→0+
lim inf
+
ξ→0
ess inf x∈D F (x, ξ)
> −∞.
ξp
Then, for each
λ ∈ Λ = 0, sup R b
r>0
a
r
,
maxt∈W ( (b−a)p−1 pM r ) F (x, t)dx
2p
problem (3.19) admits at least one nontrivial classical solution uλ ∈ W01,p ([a, b])
such that limλ→0+ kuλ k = 0 and the real function
Z b
Z b
λ 7→
H(u0λ (x))dx − λ
F (x, uλ (x))dx
a
a
is negative and strictly decreasing in the open interval Λ.
10 S. HEIDARKHANI, M. FERRARA, G. A. AFROUZI, G. CARISTI, S. MORADI
EJDE-2016/56
Now, we point out the following result, as a consequence of Theorem 3.8 in which
the function f has separated variables.
Theorem 3.9. Let α ∈ L∞ ([a, b]) with ess inf x∈[a,b] α(x) > 0. Further, let f : R →
Rξ
R be a continuous function such that f (0) = 0. Put F (ξ) = 0 f (t)dt for all ξ ∈ R,
and assume that
1
r
sup
> 1,
Rb
2 M r F (ξ)
max
(b−a)
r>0
α(x)dx
ξ∈W (
)
a
2
F (ξ)
= +∞.
lim
ξ→0+ ξ 2
Then, for each
1
r
λ ∈ Λ = 0, R b
sup
,
α(x)dx r>0 maxξ∈W ( (b−a)22 M r ) F (ξ)
a
the problem
−u00 = λα(x)f (u)h(u0 ),
x ∈ (a, b),
u(a) = u(b) = 0
(3.20)
admits at least one nontrivial classical solution uλ ∈ W01,2 ([a, b]) such that
lim kuλ k = 0
λ→0+
and the real function
Z
λ→
b
H(u0λ (x))dx
Z
−λ
a
b
α(x)F (uλ (x))dx
a
is negative and strictly decreasing in Λ.
Next we given an example to illustrate Theorem 3.9.
Example 3.10. Consider the problem
−u00 = λ(1 + ex )f (u)h(u0 ),
x ∈ (0, 1),
u(0) = u(1) = 0
where f (t) = t2 sin2 (t) − sin(t) for t ∈ R


1,
1
h(t) = 1+t
2,

1
,
2
(3.21)
and
if t ∈ (−∞, 0),
if t ∈ [0, 1],
if t ∈ (1, +∞).
We observe that f and h are continuous and h is bounded with m = inf t∈R h(t) =
and M = supt∈R h(t) = 1. By the expression of f and h we have
1
1 2
1
F (t) = t3 −
t sin(2t) + t cos(2t) − sin(2t) + cos(t) − 1
6
4
2
for every t ∈ R and

1 2

if t ∈ (−∞, 0),
2t ,
1 2
1 4
H(t) = 2 t + 12
t ,
if t ∈ [0, 1],

2 1
1
t − 2 t + 12
, if t ∈ (1, +∞).
1
2
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EXISTENCE OF SOLUTIONS
11
It is easy to see that all assumptions of Theorem 3.9 are satisfied. By Theorem 3.9
1
for each λ ∈ (0, 4e
), problem (3.21) admits at least one nontrivial classical solution
1,2
uλ in W0 ([0, 1]). Moreover, limλ→0+ kuλ k = 0 and the real function
Z 1
Z 1
0
λ 7→
H(uλ (x))dx − λ
(1 + ex )F (uλ (x))dx
0
0
is negative and strictly decreasing in (0, 1/(4e)).
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Shapour Heidarkhani
Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah,
Iran
E-mail address: s.heidarkhani@razi.ac.ir
Massimiliano Ferrara
Department of Law and Economics, University Mediterranea of Reggio Calabria, Via
dei Bianchi, 2 - 89127 Reggio Calabria, Italy
E-mail address: massimiliano.ferrara@unirc.it
12 S. HEIDARKHANI, M. FERRARA, G. A. AFROUZI, G. CARISTI, S. MORADI
EJDE-2016/56
Ghasem A. Afrouzi
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
E-mail address: afrouzi@umz.ac.ir
Giuseppe Caristi
Department of Economics, University of Messina, via dei Verdi, 75, Messina, Italy
E-mail address: gcaristi@unime.it
Shahin Moradi
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
E-mail address: shahin.moradi86@yahoo.com