Critical points approaches to elliptic problems driven by a p(x)-Laplacian
We establish the existence of at least three solutions for elliptic problems driven by a p(x)-Laplacian. The existence of at least one nontrivial solution is also proved. The approaches are based on the variational methods and critical-point theory. Встановлено існування принаймні трьох розв'яз...
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Інститут математики НАН України
2014
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| Цитувати: | Critical points approaches to elliptic problems driven by a p(x)-Laplacian / S. Heidarkhani, B. Ge // Український математичний журнал. — 2014. — Т. 66, № 12. — С. 1676–1693. — Бібліогр.: 31 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860080344477204480 |
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| author | Heidarkhani, S. Ge, B. |
| author_facet | Heidarkhani, S. Ge, B. |
| citation_txt | Critical points approaches to elliptic problems driven by a p(x)-Laplacian / S. Heidarkhani, B. Ge // Український математичний журнал. — 2014. — Т. 66, № 12. — С. 1676–1693. — Бібліогр.: 31 назв. — англ. |
| collection | DSpace DC |
| container_title | Український математичний журнал |
| description | We establish the existence of at least three solutions for elliptic problems driven by a p(x)-Laplacian. The existence of at least one nontrivial solution is also proved. The approaches are based on the variational methods and critical-point theory.
Встановлено існування принаймні трьох розв'язків еліптичних задач з p(x)-лапласіаном. Існування щонайменше одного нетривіального розв'язку також продемонстровано. Застосовані підходи ґрунтуються на варіаційних методах та теорії критичних точок.
|
| first_indexed | 2025-12-07T17:16:18Z |
| format | Article |
| fulltext |
UDC 517.9
S. Heidarkhani* (Razi Univ., Kermanshah; School Math., Inst. Res. Fundam. Sci. (IPM), Tehran, Iran),
B. Ge (Harbin Eng. Univ., China)
CRITICAL POINTS APPROACHES TO ELLIPTIC PROBLEMS
DRIVEN BY A p(x)-LAPLACIAN
ПIДХОДИ ДО ЕЛIПТИЧНИХ ЗАДАЧ З p(x)-ЛАПЛАСIАНОМ
ОСНОВАНI НА ТЕОРIЇ КРИТИЧНИХ ТОЧОК
We establish the existence of at least three solutions for elliptic problems driven by a p(x)-Laplacian. The existence of at
least one nontrivial solution is also proved. The approaches are based on variational methods and critical point theory.
Встановлено iснування принаймнi трьох розв’язкiв елiптичних задач з p(x)-лапласiаном. Iснування щонайменше
одного нетривiального розв’язку також продемонстровано. Застосованi пiдходи ґрунтуються на варiацiйних методах
та теорiї критичних точок.
1. Introduction. This paper treats the following elliptic problem:
−∆p(x)u = λf(x, u) in Ω,
u = 0 on ∂Ω,
(1)
where ∆p(x)u = div(|∇u|p(x)−2∇u) is the p(x)-Laplacian operator, Ω ⊂ RN , N ≥ 1, is a nonempty
bounded open set with a smooth boundary ∂Ω, p ∈ C(Ω) satisfies the condition N < p− :=
:= infx∈Ω p(x) ≤ p(x) ≤ supx∈Ω p(x) < +∞, λ > 0, f : Ω × R → R is an L1-Carathéodory
function.
Recently, the study of differential equations and variational problems with variable exponent has
been a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids,
etc. (see [29, 31]). It also has wide applications in different research fields, such as image processing
model (see, e.g., [16, 24]), stationary thermo-rheological viscous flows (see [1]) and the mathematical
description of the processes filtration of an idea barotropic gas through a porous medium (see [2]).
Let us point out that when p(x) = p = constant, there is a large literature which deals with
problems involving the p-Laplacian with Dirichlet boundary conditions both in the scalar case and
elliptic systems in bounded or unbounded domains, which we do not need to cite here since the reader
may easily find such papers. Many authors investigated the existence and multiplicity of solutions
for problems involving p(x)-Laplacian. In recent years there has been an increasing interest in the
study of variational problems and elliptic equations with variable exponent. We refer to [19, 21, 27]
for the theory of Lp(x) and W 1,p(x)(Ω). The case of p(x)-Laplacian with Dirichlet conditions on
the scalar case has been studied by Fan and Zhang [20]. In fact, in [20], Fan and Zhang first
introduced some basic properties of the generalized Lebesgue – Sobolev spaces W 1,p(x)
0 (Ω) which
can be regarded as a special class of generalized Orlicz – Sobolev spaces, and second they presented
several important properties of p(x)-Laplace operator, and finally under some appropriate conditions
on the nonlinear term, they established some existence results of weak solutions of the problem (1).
* Research of S. Heidarkhani was in part supported by a grant from IPM (No. 91470046).
c© S. HEIDARKHANI, B. GE, 2014
1676 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
CRITICAL POINTS APPROACHES TO ELLIPTIC PROBLEMS DRIVEN BY A p(x)-LAPLACIAN 1677
Bonanno and Chinnì in [7], employing a three critical point theorem for nondifferentiable functionals
due to Bonanno and Marano [12] (Theorem 3.6), established the existence of at least three weak
solutions for the problem
−∆p(x)u = λ(f(x, u) + µg(x, u)) in Ω,
u = 0 on ∂Ω,
where Ω ⊂ RN , N ≥ 1, is a nonempty bounded open set with a smooth boundary ∂Ω, p ∈ C(Ω),
λ and µ are two positive parameters and f, g : Ω× R→ R are two measurable with respect to each
variable separately functions and possibly discontinuous with respect to u. Bonanno and Chinnì also
in [8] based on a convenient form of the recent three critical points theorem obtained by G. Bonanno
and S. A. Marano [12] investigated multiplicity of solutions for the problem (1).
In the present paper, motivated by [7, 8], first employing two related three critical points theorems
for differentiable functionals due to Bonanno and Candito [6], we ensure the existence of at least
three weak solutions for the problem (1) (see Theorems 4 and 5) and then, using a very recent a local
minimum theorem for differentiable functionals due to Bonanno [5], under different assumptions
which have been assumed in Theorems 4 and 5 we establish the existence of at least one nontrivial
weak solution for the problem (1) (see Theorem 7). Theorems 4 and 5 extend the results of [8].
For a thorough account on the subject, we refer the reader to the papers [9, 10, 13, 14, 22, 23, 26].
2. Preliminaries and basic notations. In this section, we introduce some definitions and results
which will be used in the next section. Firstly, we introduce some theories of Lebesgue – Sobolev
spaces with variable exponent. The details can be found in [17, 19, 21]. Set
L∞+ (Ω) =
{
p ∈ L∞(Ω) : ess inf
x∈Ω
p(x) > 1
}
.
For p ∈ L∞+ (Ω), denote
p− = p−(Ω) = ess inf
x∈Ω
p(x) and p+ = p+(Ω) = ess sup
x∈Ω
p(x)
for any p(x) ∈ L∞+ (Ω), we define the variable exponent Lebesgue space
Lp(x)(Ω) =
u : u is a measurable real-valued function such that
∫
Ω
|u(x)|p(x)dx <∞
.
We define a norm, the so-called Luxemburg norm, on this space by the formula
‖u‖Lp(x)(Ω) = inf
λ > 0 :
∫
Ω
∣∣∣∣u(x)
λ
∣∣∣∣p(x)
dx ≤ 1
.
The space (Lp(x)(Ω), ‖.‖p(x)) is a Banach space. Define the variable exponent Sobolev space
W 1,p(x)(Ω) by
W 1,p(x)(Ω) =
{
u ∈ Lp(x)(Ω) : |∇u| ∈ Lp(x)(Ω)
}
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
1678 S. HEIDARKHANI, B. GE
equipped with the norm
‖u‖W 1,p(x)(Ω) = ‖u‖Lp(x)(Ω) + ‖∇u‖Lp(x)(Ω).
Denote by W 1,p(x)
0 (Ω) the closure of C∞0 (Ω) in W 1,p(x)(Ω). On W 1,p(x)
0 (Ω) we consider the norm
‖u‖ : = ‖∇u‖Lp(x)(Ω).
Here we display some facts which will be used later.
Proposition 1 (see [20]). (i) The spaces Lp(x)(Ω), W 1,p(x)(Ω) and W 1,p(x)
0 (Ω) are separable
and reflexive Banach spaces.
(ii) There is a constant c > 0 such that ‖u‖Lp(x)(Ω) ≤ c‖∇u‖Lp(x)(Ω) for all u ∈W 1,p(x)
0 (Ω).
Proposition 2 (see [7]). Set ρp(u) =
∫
Ω
|u(x)|p(x)dx. For u ∈W 1,p(x)
0 (Ω), we have
(i) ‖u‖ < 1(= 1;> 1)⇐⇒ ρp(|∇u|) < 1(= 1;> 1).
(ii) If ‖u‖ > 1, then
1
p+
‖u‖p− ≤
∫
Ω
1
p(x)
|∇u(x)|p(x)dx ≤ 1
p−
‖u‖p+ .
(iii) If ‖u‖ < 1, then
1
p+
‖u‖p+ ≤
∫
Ω
1
p(x)
|∇u(x)|p(x)dx ≤ 1
p−
‖u‖p− .
As pointed in [20, 27], W 1,p(x)(Ω) is continuously embedded in W 1,p−(Ω) and, since p− > N,
W 1,p−(Ω) is compactly embedded in C0(Ω). Thus W 1,p(x)(Ω) is compactly embedded in C0(Ω).
So, in particular, there exists a positive constant c0 such that
‖u‖C0(Ω) ≤ c0‖u‖ (2)
for each u ∈W 1,p(x)
0 (Ω).
Let X denotes the Sobolev space W 1,p(x)
0 (Ω). Let G(u) =
∫
Ω
1
p(x)
|∇u(x)|p(x)dx for all u ∈ X.
We denote L = G′ : X → X∗, then
L(u)(v) =
∫
Ω
|∇u(x)|p(x)−2∇u(x)∇v(x) dx
for all u, v ∈ X.
Proposition 3 (see [20]). (i) L : X → X∗ is a continuous, bounded and strictly monotone
operator.
(ii) L is a mapping of type (S+), i.e., if un ⇀ u in X and lim supn→∞(L(un), un − u) ≤ 0,
then un → u in X.
(iii) L : X → X∗ is a homeomorphism.
We say that u is a weak solution to the problem (1) if u ∈ X and∫
Ω
|∇u(x)|p(x)−2∇u(x)∇v(x)dx− λ
∫
Ω
f(x, u(x))v(x)dx = 0
for every v ∈ X.
Put
δ(x) = sup
{
δ > 0 : S(x, δ) ⊂ Ω
}
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
CRITICAL POINTS APPROACHES TO ELLIPTIC PROBLEMS DRIVEN BY A p(x)-LAPLACIAN 1679
where S(x, δ) denotes the ball with center at x and radius of δ, for all x ∈ Ω, one can prove that
there exists x0 ∈ Ω such that S(x0, D) ⊂ Ω, where D = supx∈Ω δ(x). Set
m :=
πN/2
N
2
Γ
(
N
2
)
where Γ is the Euler function, and for each r > 0, set
γr := max
{
(p+r)1/p− , (p+r)1/p+
}
.
Put
F (x, t) =
t∫
0
f(x, ξ) dξ
for all (x, t) ∈ Ω×R.
3. Existence of three solutions. In the following section we establish the existence of at least
three weak solutions for the problem (1). Our main tools are two three critical points theorems. In
the first one the coercivity of the functional Φ − λΨ is required, in the second one a suitable sign
hypothesis is assumed. The first result has been obtained in [4], the second one in [3]. Here we recall
them as given in [6].
Theorem 1 ([6], Theorem 3.2). Let X be a reflexive real Banach space, Φ : X −→ R be a
coercive and continuously Gâteaux differentiable functional whose derivative admits a continuous
inverse on X∗, Ψ : X −→ R be a continuously Gâteaux differentiable functional whose derivative is
compact, such that
inf
X
Φ = Φ(0) = Ψ(0) = 0.
Assume that there is a positive constant r and v ∈ X, with 2r < Φ(v), such that
(a1)
supu∈Φ−1(]−∞,r[) Ψ(u)
r
<
2
3
Ψ(v)
Φ(v)
;
(a2) for all λ ∈
]
3
2
Φ(v)
Ψ(v)
,
r
supu∈Φ−1(]−∞,r[) Ψ(u)
[
, the functional Φ− λΨ is coercive.
Then, for each λ ∈
]
3
2
Φ(v)
Ψ(v)
,
r
supu∈Φ−1(]−∞,r[) Ψ(u)
[
the functional Φ − λΨ has at least three
distinct critical points.
Theorem 2 ([6], Theorem 3.3). Let X be a reflexive real Banach space, Φ : X −→ R be a
convex, coercive and continuously Gâteaux differentiable functional whose derivative admits a con-
tinuous inverse on X∗, Ψ : X −→ R be a continuously Gâteaux differentiable functional whose
derivative is compact, such that
1) infX Φ = Φ(0) = Ψ(0) = 0;
2) for each λ > 0 and for every u1, u2 which are local minimum for the functional Φ− λΨ and
such that Ψ(u1) ≥ 0 and Ψ(u2) ≥ 0, one has
inf
s∈[0,1]
Ψ(su1 + (1− s)u2) ≥ 0.
Assume that there are two positive constants r1, r2 and v ∈ X, with 2r1 < Φ(v) <
r2
2
, such that
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
1680 S. HEIDARKHANI, B. GE
(b1)
supu∈Φ−1(]−∞,r1[) Ψ(u)
r1
<
2
3
Ψ(v)
Φ(v)
;
(b2)
supu∈Φ−1(]−∞,r2[) Ψ(u)
r2
<
1
3
Ψ(v)
Φ(v)
.
Then, for each λ ∈
]
3
2
Φ(v)
Ψ(v)
, min
{
r1
supu∈Φ−1(]−∞,r1[) Ψ(u)
,
r2/2
supu∈Φ−1(]−∞,r2[) Ψ(u)
}[
, the func-
tional Φ− λΨ has at least three distinct critical points which lie in Φ−1(]−∞, r2[).
A special case of our main results is the following theorem.
Theorem 3. Let Ω ⊆ R2 be a nonempty bounded open set with a smooth boundary ∂Ω. Let f :
R → R be a continuous function and put F (t) =
∫ t
0
f(ξ) dξ for all t ∈ R such that F (h) > 0 for
some h > 0 and F (ξ) ≥ 0 in [0, h]. Fix p(x) = p > 2 and assume that
lim inf
ξ→0
F (ξ)
|ξ|p
= lim sup
|ξ|→+∞
F (ξ)
|ξ|p
= 0.
Then, there is λ∗ > 0 such that for each λ > λ∗ the problem
−∆pu = λf(u) in Ω,
u = 0 on ∂Ω
admits at least three weak solutions.
Remark 1. Similar results to Theorem 3 have been obtained in [11] (Theorem 0) in which a class
of Dirichlet quasilinear elliptic systems driven by a (p, q)-Laplacian operator has been considered, and
also in [25] (Theorem 1) in which a quasilinear second-order differential equation has been studied.
We formulate the existence results as follows:
Theorem 4. Let f : Ω×R→ R be an L1-Carathéodory function such that ess infx∈Ω F (x, ξ) ≥
≥ 0 for all ξ ∈ R. Assume that there exist two positive constants r and h such that
(A1)
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
> 2r;
(A2)
∫
Ω
sup|t|≤c0γr F (x, t)dx
r
<
2 ess infx∈Ω F (x, h)
3
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
(2N − 1)
;
(A3) lim sup|t|→+∞
F (x, t)
|t|p−/p+
<
∫
Ω
sup|t|≤c0γr F (x, t)dx
r
.
Then, for each
λ ∈
3
2
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
(2N − 1)
ess infx∈Ω F (x, h)
,
r∫
Ω
sup|t|≤c0γr F (x, t)dx
the problem (1) admits at least three weak solutions.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
CRITICAL POINTS APPROACHES TO ELLIPTIC PROBLEMS DRIVEN BY A p(x)-LAPLACIAN 1681
Proof. In order to apply Theorem 1 to our problem, we introduce the functionals Φ, Ψ :
X → R for each u ∈ X, as follows:
Φ(u) =
∫
Ω
1
p(x)
∣∣∇u(x)
∣∣p(x)
dx
and
Ψ(u) =
∫
Ω
F (x, u(x))dx.
It is well known that Φ and Ψ are well defined and continuously differentiable functionals whose
derivatives at the point u ∈ X are the functionals Φ′(u),Ψ′(u) ∈ X∗, given by
Φ′(u)(v) =
∫
Ω
∣∣∇u(x)
∣∣p(x)−2∇u(x)∇v(x)dx
and
Ψ′(u)(v) =
∫
Ω
f(x, u(x))v(x)dx
for every v ∈ X, respectively, as well as Ψ is sequentially weakly upper semicontinuous. Moreover, Φ
is sequentially weakly lower semicontinuous and Φ′ admits a continuous inverse on X∗. Furthermore,
Ψ′ : X → X∗ is a compact operator. Set
w(x) =
0 if x ∈ Ω \ S(x0, D),
h if x ∈ S
(
x0,
D
2
)
,
2h
D
(
D −
√∑N
i=1
(xi − x0i)
2
)
if x ∈ S(x0, D) \ S
(
x0,
D
2
)
.
(3)
It is easy to see that w ∈ X and, in particular, one has
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
≤ Φ(w) ≤
≤ 1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
(4)
and
Ψ(w) ≥
∫
S(x0, D/2)
F (x,w(x))dx ≥ ess inf
x∈Ω
F (x, h)m
(
D
2
)N
. (5)
From (A1), taking (4) into account, we get Φ(w) > 2r. Thanks to the embedding X ↪→ C0(Ω), we
have
Φ−1(]−∞, r[) = {u ∈ X; Φ(u) < r} =
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
1682 S. HEIDARKHANI, B. GE
=
u ∈ X;
∫
Ω
1
p(x)
|∇u(x)|p(x)dx < r
⊆
⊆
{
u ∈ X; |u(x)| ≤ c0γr for all x ∈ Ω
}
,
and it follows that
sup
u∈Φ−1(]−∞,r[)
Ψ(u) = sup
u∈Φ−1(]−∞,r[)
∫
Ω
F (x, u(x))dx ≤
≤
∫
Ω
sup
|t|≤c0γr
F (x, t) dx.
Therefore, owing to the assumption (A2), (4) and (5), we get
supu∈Φ−1(]−∞,r[) Ψ(u)
r
=
supu∈Φ−1(]−∞,r[)
∫
Ω
F (x, u(x))dx
r
≤
≤
∫
Ω
sup|t|≤c0γr F (x, t)dx
r
<
<
2
3
ess infx∈Ω F (x, h)
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
(2N − 1)
≤ 2
3
Ψ(w)
Φ(w)
.
Furthermore, from (A3) there exist two constants η, ϑ ∈ R with
η <
∫
Ω
sup|t|≤c0γr F (x, t)dx
r
such that
|Ω|c0F (x, t) ≤ η |t|
p−
p+
+ ϑ for all x ∈ Ω and for all t ∈ Rn.
Fix u ∈ X. Then
F (x, u(x)) ≤ 1
|Ω|c0
(
η
|u(x)|p−
p+
+ ϑ
)
for all x ∈ Ω. (6)
Now, in order to prove the coercivity of the functional Φ − λΨ, first we assume that η > 0. So, if
‖u‖ ≥ 1, for any fixed
λ ∈
3
2
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
(2N − 1)
ess infx∈Ω F (x, h)
,
r∫
Ω
sup|t|≤c0γr F (x, t)dx
,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
CRITICAL POINTS APPROACHES TO ELLIPTIC PROBLEMS DRIVEN BY A p(x)-LAPLACIAN 1683
bearing (2) in mind, from Proposition 2 and (6) we obtain
Φ(u)− λΨ(u) =
∫
Ω
1
p(x)
|∇u(x)|p(x)dx− λ
∫
Ω
F (x, u(x))dx ≥
≥ 1
p+
‖u‖p− − λη
|Ω|c0
∫
Ω
|u(x)|p−dx
p+
− λϑ
c0
≥
1
p+
‖u‖p− − λη
|Ω|c0
|Ω|c0‖u‖p
−
p+
− λϑ
c0
≥
≥
1− η r∫
Ω
sup|t|≤c0γr F (x, t)dx
1
p+
‖u‖p− − λϑ
c0
,
and thus
lim
‖u‖→+∞
(
Φ(u)− λΨ(u)
)
= +∞.
On the other hand, if η ≤ 0. Clearly, we get lim‖u‖→+∞(Φ(u)− λΨ(u)) = +∞. Both cases lead to
the coercivity of functional Φ− λΨ.
So, the assumptions (a1) and (a2) in Theorem 1 are satisfied. Hence, by using Theorem 1, taking
into account that the weak solutions of the problem (1) are exactly the solutions of the equation
Φ′(u)− λΨ′(u) = 0, we have the conclusion.
Theorem 4 is proved.
Theorem 5. Let f : Ω×R→ R be an L1-Carathéodory function such that ess infx∈Ω F (x, ξ) ≥
≥ 0 for all ξ ∈ R and satisfies the condition f(x, t) ≥ 0 for all (x, t) ∈ Ω × (R+ ∪ {0}). Assume
that there exist three positive constants r1, r2 and h with
2r1 <
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
and
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
<
r2
2
such that
(B1)
∫
Ω
sup|t|≤c0γr F (x, t)dx
r1
<
2 ess infx∈Ω F (x, h)
3
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
(2N − 1)
;
(B2)
∫
Ω
sup|t|≤c0γr F (x, t)dx
r2
<
ess infx∈Ω F (x, h)
3
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
(2N − 1)
.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
1684 S. HEIDARKHANI, B. GE
Then, for each
λ ∈
3
2
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
(2N − 1)
ess infx∈Ω F (x, h)
,
min
r1∫
Ω
sup|t|≤c0γr F (x, t)dx
,
r2
2∫
Ω
sup|t|≤c0γr F (x, t)dx
the problem (1) admits at least three nonnegative weak solutions v1, v2, v3 such that∫
Ω
1
p(x)
|∇vj(x)|p(x)dx ≤ r2 for each x ∈ Ω, j = 1, 2, 3.
Proof. Let Φ and Ψ be as in the proof of Theorem 4. Let us employ Theorem 2 to our functionals.
Obviously, Φ and Ψ satisfy the condition 1 of Theorem 2. Now, we verify that the functional Φ−λΨ
satisfies the assumption 2 of Theorem 2. Let u? and u?? be two local minima for Φ− λΨ. Then u?
and u?? are critical points for Φ − λΨ, and so, they are weak solutions for the problem (1). Since
f(x, t) ≥ 0 for all (x, t) ∈ Ω × (R+ ∪ {0}), from the Weak Maximum Principle (see for instance
[15]) we deduce u?(x) ≥ 0 and u??(x) ≥ 0 for every x ∈ Ω. So, it follows that su?+ (1− s)u?? ≥ 0
for all s ∈ [0, 1], and that f(su? + (1 − s)u??, t) ≥ 0, and consequently, Ψ(su? + (1 − s)u??) ≥ 0
for all s ∈ [0, 1]. Moreover, from the conditions
2r1 <
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
and
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
<
r2
2
,
we observe 2r1 < Φ(w) <
r2
2
. Next, thanks to the embedding X ↪→ C0(Ω), we have
Φ−1(]−∞, r1[) = {u ∈ X; Φ(u) < r1} =
=
u ∈ X;
∫
Ω
1
p(x)
|∇u(x)|p(x)dx < r1
⊆
⊆
{
u ∈ X; |u(x)| ≤ c0γr1 for all x ∈ Ω
}
,
and it follows that
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CRITICAL POINTS APPROACHES TO ELLIPTIC PROBLEMS DRIVEN BY A p(x)-LAPLACIAN 1685
sup
u∈Φ−1(]−∞,r1[)
Ψ(u) = sup
u∈Φ−1(]−∞,r1[)
∫
Ω
F (x, u(x))dx ≤
∫
Ω
sup
|t|≤c0γr
F (x, t)dx.
Therefore, owing to the assumption (B1), we get
supu∈Φ−1(]−∞,r1[) Ψ(u)
r1
=
supu∈Φ−1(]−∞,r1[)
∫
Ω
F (x, u(x))dx
r1
≤
≤
∫
Ω
sup|t|≤c0γr1 F (x, t)dx
r1
<
<
2
3
ess infx∈Ω F (x, h)
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
(2N − 1)
≤ 2
3
Ψ(w)
Φ(w)
.
As above, using the assumption (B2), we obtain
supu∈Φ−1(]−∞,r2[) Ψ(u)
r2
=
supu∈Φ−1(]−∞,r2[)
∫
Ω
F (x, u(x))dx
r2
≤
≤
∫
Ω
sup|t|≤c0γr2 F (x, t)dx
r2
<
<
1
3
ess infx∈Ω F (x, h)
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
(2N − 1)
≤ 1
3
Ψ(w)
Φ(w)
.
So, the assumptions (b1) and (b2) in Theorem 2 are satisfied. Hence, by using Theorem 2, taking
into account that the weak solutions of the problem (1) are exactly the solutions of the equation
Φ′(u)− λΨ′(u) = 0, the problem (1) admits at least three distinct weak solutions in X.
Theorem 5 is proved.
We end this section by proving Theorem 3.
Proof of Theorem 3. Fix
λ > λ∗ :=
9
2
1
p
(
2h
D
)p
F (h)
.
Taking into account that lim infξ→0
F (ξ)
|ξ|p
= 0 there is {rn}n∈N ⊆ ]0,+∞[ such that limn→+∞ rn =
= 0 and
lim
n→+∞
|Ω|max|t|≤c0(prn)1/p F (t)
rn
= 0.
Hence, there is r > 0 such that
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
1686 S. HEIDARKHANI, B. GE
|Ω|max|t|≤c0(pr)1/p F (t)
r
< min
2p
9
F (h)(
2h
D
)p ;
1
λ
and 2r <
3D2π
4p
(
2h
D
)p
.
From Theorem 4 the conclusion follows.
4. Existence of a nontrivial solution. First we here recall for the reader’s convenience [28]
(Theorem 2.5) as given in [5] (Theorem 5.1) (see also [5] (Proposition 2.1) for related results) which
is our main tool to prove the main result.
For a given nonempty set X, and two functionals Φ,Ψ : X → R, we define the following
functions:
ϑ(r1, r2) = inf
v∈Φ−1(]r1,r2[)
supu∈Φ−1(]r1,r2[) Ψ(u)−Ψ(v)
r2 − Φ(v)
and
ρ(r1, r2) = sup
v∈Φ−1(]r1,r2[)
Ψ(v)− supu∈Φ−1(]−∞,r1[) Ψ(u)
Φ(v)− r1
for all r1, r2 ∈ R, r1 < r2.
Theorem 6 ([5], Theorem 5.1). Let X be a reflexive real Banach space, Φ : X → R be a se-
quentially weakly lower semicontinuous, coercive and continuously Gâteaux differentiable functional
whose Gâteaux derivative admits a continuous inverse on X∗ and Ψ : X → R be a continuously
Gâteaux differentiable functional whose Gâteaux derivative is compact. Put Iλ = Φ−λΨ and assume
that there are r1, r2 ∈ R, r1 < r2, such that
ϑ(r1, r2) < ρ(r1, r2).
Then, for each λ ∈
]
1
ρ(r1, r2)
,
1
ϑ(r1, r2)
[
there is u0,λ ∈ Φ−1(]r1, r2[) such that Iλ(u0,λ) ≤ Iλ(u)
∀u ∈ Φ−1(]r1, r2[) and I ′λ(u0,λ) = 0.
We formulate the main result of this section as follows:
Theorem 7. Let f : Ω×R→ R be an L1-Carathéodory function such that ess infx∈Ω F (x, ξ) ≥
≥ 0 for all ξ ∈ R. Assume that there exist a nonnegative constant r1 and two positive constants r2
and h with
r1 <
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
and
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
< r2
such that
(C1)
∫
Ω
sup|t|≤c0γr2 F (x, t)dx− ess infx∈Ω F (x, h)m
(
D
2
)N
r2 −
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+} <
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CRITICAL POINTS APPROACHES TO ELLIPTIC PROBLEMS DRIVEN BY A p(x)-LAPLACIAN 1687
<
∫
Ω
sup|t|≤c0γr1 F (x, t)dx− ess infx∈Ω F (x, h)m
(
D
2
)N
r1 −
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+} . (7)
Then, for each
λ ∈
r1 −
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
∫
Ω
sup|t|≤c0γr1 F (x, t)dx− ess infx∈Ω F (x, h)m
(
D
2
)N ,
r2 −
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
∫
Ω
sup|t|≤c0γr2 F (x, t)dx− ess infx∈Ω F (x, h)m
(
D
2
)N
the problem (1) admits at least one nontrivial weak solution u0 ∈ X such that
r1 <
∫
Ω
1
p(x)
|∇u0(x)|p(x)dx < r2.
Proof. In order to apply Theorem 6 to our problem, let the functionals Φ,Ψ : X → R be as in the
proof of Theorem 4. As seen in the proof of Theorem 4, Φ and Ψ satisfy the regularity assumptions
of Theorem 6. Choose w as given in (3). Taking (4) into account, from the conditions
r1 <
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
and
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
< r2
we get
r1 < Φ(w) < r2.
Thanks to the embedding X ↪→ C0(Ω), we have
Φ−1(]−∞, r2[) = {u ∈ X; Φ(u) < r2} =
=
u ∈ X;
∫
Ω
1
p(x)
|∇u(x)|p(x)dx < r2
⊆
⊆
{
u ∈ X; |u(x)| ≤ c0γr2 for all x ∈ Ω
}
,
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1688 S. HEIDARKHANI, B. GE
and it follows that
sup
u∈Φ−1(]−∞,r2[)
Ψ(u) = sup
u∈Φ−1(]−∞,r2[)
∫
Ω
F (x, u(x))dx ≤
∫
Ω
sup
|t|≤c0γr2
F (x, t)dx.
Therefore, one has
ϑ(r1, r2) ≤
supu∈Φ−1(]−∞,r2[) Ψ(u)−Ψ(w)
r2 − Φ(w)
≤
≤
∫
Ω
sup|t|≤c0γr2 F (x, t)dx−Ψ(w)
r2 − Φ(w)
≤
≤
∫
Ω
sup|t|≤c0γr2 F (x, t)dx− ess infx∈Ω F (x, h)m
(
D
2
)N
r2 −
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+} .
On the other hand, arguing as before, one has
ρ(r1, r2) ≥
Ψ(w)− supu∈Φ−1(]−∞,r1[) Ψ(u)
Φ(w)− r1
≥
≥
Ψ(w)−
∫
Ω
sup|t|≤c0γr1 F (x, t)dx
Φ(w)− r1
≥
≥
∫
Ω
sup|t|≤c0γr1 F (x, t)dx− ess infx∈Ω F (x, h)m
(
D
2
)N
r1 −
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+} .
Hence, from Assumption (C1), one has ϑ(r1, r2) < ρ(r1, r2). Therefore, applying Theorem 6, for
each
λ ∈
r1 −
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
∫
Ω
sup|t|≤c0γr1 F (x, t)dx− ess infx∈Ω F (x, h)m
(
D
2
)N ,
r2 −
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
∫
Ω
sup|t|≤c0γr2 F (x, t)dx− ess infx∈Ω F (x, h)m
(
D
2
)N
,
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CRITICAL POINTS APPROACHES TO ELLIPTIC PROBLEMS DRIVEN BY A p(x)-LAPLACIAN 1689
the functional Φ − λΨ has admits at least one critical point u0 ∈ X such that r1 < Φ(u0) < r2,
that is
r1 <
∫
Ω
1
p(x)
|∇u0(x)|p(x)dx < r2.
Hence, taking into account that the weak solutions of the problem (1) are exactly the solutions of the
equation Φ′(u)− λΨ′(u) = 0, we achieve the stated assertion.
Theorem 7 is proved.
Now we point out the following consequence of Theorem 7.
Theorem 8. Suppose that
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
≤ 1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
.
Let f : Ω×R→ R be an L1-Carathéodory function such that ess infx∈Ω F (x, ξ) ≥ 0 for all ξ ∈ R.
Assume that there exist two positive constants r and h with
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
< r
such that
(C2)
∫
Ω
sup|t|≤c0γr F (x, t)dx
r
<
ess infx∈Ω F (x, h)m
(
D
2
)N
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+} .
Then, for each
λ ∈
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
ess infx∈Ω F (x, h)m
(
D
2
)N ,
r∫
Ω
sup|t|≤c0γr F (x, t)dx
the problem (1) admits at least one nontrivial weak solution u0 ∈ X such that
r1 <
∫
Ω
1
p(x)
|∇u0(x)|p(x)dx < r2.
Proof. The conclusion follows from Theorem 7 by taking r1 = 0 and r2 = r. Indeed, owing to
our assumptions, one has∫
Ω
sup|t|≤c0γr F (x, t)dx− ess infx∈Ω F (x, h)m
(
D
2
)N
r − 1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+} <
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
1690 S. HEIDARKHANI, B. GE
<
(
1−
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
r
)∫
Ω
sup|t|≤c0γr F (x, t)dx
r − 1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+} =
=
∫
Ω
sup|t|≤c0γr F (x, t)dx
r
<
ess infx∈Ω F (x, h)m
(
D
2
)N
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+} ≤
≤
ess infx∈Ω F (x, h)m
(
D
2
)N
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+} .
In particular, one has∫
Ω
sup|t|≤c0γr F (x, t)dx− ess infx∈Ω F (x, h)m
(
D
2
)N
r − 1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+} <
∫
Ω
sup|t|≤c0γr F (x, t)dx
r
.
Hence, Theorem 7 concludes the result.
Theorem 8 is proved.
Let f : R→ R be a continuous function, and put F (t) =
∫ t
0
f(ξ) dξ for all t ∈ R. We have the
following result as a direct consequence of Theorem 7.
Theorem 9. Let f : R → R be a nonnegative continuous function. Assume that there exist a
nonnegative constant r1 and two positive constants r2 and h with
r1 <
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
and
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
mDN 2N − 1
2N
< r2
such that
(C3)
|Ω|F (c0γr2)− F (h)m
(
D
2
)N
r2 −
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+} <
|Ω|F (c0γr1)− F (h)m
(
D
2
)N
r1 −
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+} .
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CRITICAL POINTS APPROACHES TO ELLIPTIC PROBLEMS DRIVEN BY A p(x)-LAPLACIAN 1691
Then, for each
λ ∈
r1 −
1
p+
min
{(
2h
D
)p−
,
(
2h
D
)p+}
|Ω|F (c0γr1)− F (h)m
(
D
2
)N ,
r2 −
1
p−
max
{(
2h
D
)p−
,
(
2h
D
)p+}
|Ω|F (c0γr2)− F (h)m
(
D
2
)N
the problem
−∆p(x)u = λf(u) in Ω,
u = 0 on ∂Ω
admits at least one nontrivial weak solution u0 ∈ X such that
r1 <
∫
Ω
1
p(x)
|∇u0(x)|p(x)dx < r2.
We end this paper by giving the following special case of our main result of this section.
Theorem 10. Let p(x) = p > N. Let h : Ω→ R be a positive and essentially bounded function
and g : R→ R be a nonnegative function such that
lim
t→0+
g(t)
tp−1
= +∞.
Then, for each λ ∈
0,
1∫
Ω
h(x)dx
supr>0
r∫ c0(pr)1/p
0
g(ξ) dξ
, the problem
−∆pu = λh(x)g(u) in Ω,
u = 0 on ∂Ω
admits at least one nontrivial weak solution in X.
Proof. For fixed λ ∈
0,
1∫
Ω
h(x)dx
supr>0
r∫ c0(pr)1/p
0
g(ξ) dξ
, there exists positive
constant r such that
λ <
1∫
Ω
h(x)dx
r∫ c0(pr)1/p
0
g(ξ)dξ
.
Moreover, the condition limt→0+
g(t)
tp−1
= +∞ implies limt→0+
∫ t
0
g(ξ)dξ
tp = +∞. Therefore, we
can choose positive constant h satisfying
1
p
(
2h
D
)p
mDN 2N − 1
2N
< r such that
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
1692 S. HEIDARKHANI, B. GE
(
1
λ
)
2p
pDp ess infx∈Ω h(x)m
(
D
2
)N <
∫ h
0
g(ξ)dξ
hp
.
Hence, Theorem 8 ensures the conclusion.
Remark 2. All proofs in this paper are based on the computation of value of the corresponding
functionals on the function w(x) introduced in (3), which has been taken as the same as in [7].
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ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
CRITICAL POINTS APPROACHES TO ELLIPTIC PROBLEMS DRIVEN BY A p(x)-LAPLACIAN 1693
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| id | nasplib_isofts_kiev_ua-123456789-166319 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1027-3190 |
| language | English |
| last_indexed | 2025-12-07T17:16:18Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Heidarkhani, S. Ge, B. 2020-02-18T17:51:25Z 2020-02-18T17:51:25Z 2014 Critical points approaches to elliptic problems driven by a p(x)-Laplacian / S. Heidarkhani, B. Ge // Український математичний журнал. — 2014. — Т. 66, № 12. — С. 1676–1693. — Бібліогр.: 31 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/166319 517.9 We establish the existence of at least three solutions for elliptic problems driven by a p(x)-Laplacian. The existence of at least one nontrivial solution is also proved. The approaches are based on the variational methods and critical-point theory. Встановлено існування принаймні трьох розв'язків еліптичних задач з p(x)-лапласіаном. Існування щонайменше одного нетривіального розв'язку також продемонстровано. Застосовані підходи ґрунтуються на варіаційних методах та теорії критичних точок. Research of S. Heidarkhani was in part supported by a grant from IPM (No. 91470046) en Інститут математики НАН України Український математичний журнал Статті Critical points approaches to elliptic problems driven by a p(x)-Laplacian Підходи до еліптичних задач з p(x)-лапласіаном основане на теорії критичних точок Article published earlier |
| spellingShingle | Critical points approaches to elliptic problems driven by a p(x)-Laplacian Heidarkhani, S. Ge, B. Статті |
| title | Critical points approaches to elliptic problems driven by a p(x)-Laplacian |
| title_alt | Підходи до еліптичних задач з p(x)-лапласіаном основане на теорії критичних точок |
| title_full | Critical points approaches to elliptic problems driven by a p(x)-Laplacian |
| title_fullStr | Critical points approaches to elliptic problems driven by a p(x)-Laplacian |
| title_full_unstemmed | Critical points approaches to elliptic problems driven by a p(x)-Laplacian |
| title_short | Critical points approaches to elliptic problems driven by a p(x)-Laplacian |
| title_sort | critical points approaches to elliptic problems driven by a p(x)-laplacian |
| topic | Статті |
| topic_facet | Статті |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/166319 |
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