By proving an embedding theorem involving the critical exponent of anisotropic type, the authors obtained some existence and nonexistence results in the case when p>p+=max{p1,p2,.. In Se
Trang 1Contents lists available atScienceDirect Nonlinear Analysis: Real World Applications
journal homepage:www.elsevier.com/locate/nonrwa
On a class of anisotropic elliptic equations without
Ambrosetti–Rabinowitz type conditions
aDepartment of Science Management and International Cooperation, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi,
Quang Binh, Viet Nam
bDepartment of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam
a r t i c l e i n f o
Article history:
Received 8 November 2012
Accepted 24 September 2013
a b s t r a c t This article investigates a class of anisotropic elliptic equations with non-standard growth conditions
−
N
i= 1
∂x i
|∂x i u| p i(x)− 2∂x i u=f(x,u) inΩ,
whereΩ ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary∂Ω, and p i,
i=1,2, ,N are continuous functions onΩsuch that 1<p i(x) <N Using variational
methods, we obtain some existence and multiplicity results for such problems without Ambrosetti–Rabinowitz type conditions
© 2013 Elsevier Ltd All rights reserved
1 Introduction
In this paper, we are interested in the existence of solutions for the elliptic anisotropic problem with non-standard growth conditions
−
N
i= 1
∂x i
| ∂x i u|p i(x)− 2∂x i u
=f(x,u) inΩ,
(1.1)
whereΩ ⊂RN (N ≥3) is a bounded domain with smooth boundary∂Ω, and p i , i=1,2, ,N are continuous functions
onΩsuch that 1<p i(x) <N, and f :Ω×R→R is a Carathéodory function.
In the case when p i(x) =p(x)for any i=1,2, ,N, the operator involved in(1.1)has similar properties to the p(x) -Laplace operator, i.e.,∆p(x)u := div(|∇u|p(x)− 2∇u) This differential operator is a natural generalization of the isotropic
p-Laplace operator∆p u:=div(|∇u|p−2∇u), where p>1 is a real constant However, the p(x)-Laplace operator possesses
more complicated nonlinearities than the p-Laplace operator, due to the fact that∆p(x)is not homogeneous The study of
nonlinear elliptic problems (equations and systems) involving quasilinear homogeneous type operators like the p-Laplace operator is based on the theory of standard Sobolev spaces W k,p(Ω)in order to find weak solutions These spaces consist
of functions that have weak derivatives and satisfy certain integrability conditions In the case of nonhomogeneous p(x) -Laplace operators the natural setting for this approach is the use of the variable exponent Sobolev spaces Differential and
∗Corresponding author Tel.: +84 523863245.
E-mail addresses:ntchung82@yahoo.com (N.T Chung), hq_toan@yahoo.com (H.Q Toan).
1468-1218/$ – see front matter © 2013 Elsevier Ltd All rights reserved.
Trang 2partial differential equations with nonstandard growth conditions have received specific attention in recent decades The interest played by such growth conditions in elastic mechanics and electrorheological fluid dynamics has been highlighted
in many physical and mathematical works We refer to some interesting works [1–8]
In a recent paper [9], I Fragalà et al have studied the following anisotropic quasilinear elliptic problem
−
N
i= 1
∂x i
| ∂x i u|p i− 2∂x i u
= λu p−1 inΩ,
(1.2)
whereΩ ⊂ RN (N ≥3) is a bounded domain with smooth boundary∂Ω, p i >1 for all i=1,2, ,N and p >1 Note
that if p i =2 for all i=1,2, ,N then problem(1.2)reduces to the well-known semilinear equation−∆u= λu p− 1 By proving an embedding theorem involving the critical exponent of anisotropic type, the authors obtained some existence and
nonexistence results in the case when p>p+=max{p1,p2, ,p N}or p<p−=min{p1,p2, ,p N} The results in [9] have been extended by A.D Castro et al [10], in which the authors studied problem(1.2)in the case when p−<p<p+ In order to study the existence of solutions for(1.2)the above authors have found the solutions in the space W1, − →p
0 (Ω)which
is defined as the closure of C∞
0 (Ω)with respect to the norm
∥u∥− →
p =
N
i= 1
| ∂x i u|p i,
where− →p = (p1,p2, ,p N)and| |p i denotes the norm in L p i(Ω)for all i=1,2, ,N.
Since then, many authors have been interested in the existence of solutions for elliptic problems in this direction, we refer to some interesting works [11,14–19] In [14,16], the authors studied problem(1.1)in the case when the nonlinearity f verifies
the Ambrosetti–Rabinowitz type conditions ((A–R) type conditions for short), that is, there exists a positive constantµ >P+
such that
0< µF(x,t) := µ
t
0
for all x∈Ωand|t| >M>0, which implies that for some positive constants c,d, we have
F(x,t) ≥c|t|µ−d for all t∈
This says that f(x,t)is P+-superlinear at infinity in the sense that
lim
|t|→∞
F(x,t)
|t|P+
= +∞ uniformly in x∈Ω.
In [17], the author considered problem(1.1)with a particular nonlinearity f(x,t) = tα(x)− 1−tβ(x)− 1, t ≥ 0 and x ∈ Ω The functionsα(x)andβ(x)were assumed to satisfy the condition 1< β−≤ β+< α−≤ α+<P−≤P+<P− ,∞ This
means that f(x,t)is P+-sublinear at infinity Using the minimum principle combined with the mountain pass theorem, the author obtained the existence of at least two nonnegative nontrivial weak solutions The result of [17] was improved in [18],
in which the author assumed that (see the condition(10)in Theorem 2 of [18]):
max
lim sup
|t|→ 0
sup
x∈ ΩF(x,t)
|t|P+ ,lim sup
|t|→+∞
sup
x∈ ΩF(x,t)
|t|P−
≤0.
Using the three critical points theorem by B Ricceri [20] the author obtained a multiplicity result for(1.1) Regarding the problem(1.1)with Neumann boundary conditions, we refer to the papers [15,19]
In this paper, we consider problem(1.1)in the case when the nonlinear term f(x,t)is P+-superlinear at infinity but does not satisfy the (A–R) type condition(1.3)as in [14,16] More precisely, motivated by the ideas firstly introduced by O.H Miyagaki et al [21] and developed by G Li et al [22], C Ji [23], the goal this paper is to prove some existence and multiplicity results for problem(1.1)in the anisotropic case To overcome the difficulties brought, we will use the mountain pass theorem in [24] and the fountain theorem in [25] with the(C )condition (seeDefinition 3.4)
Trang 3The remainder of the paper is organized as follows In Section2, we will recall the definitions and some properties of anisotropic variable exponent Sobolev spaces The readers can consult the paper [11–13] for details on this class of functional spaces In Section3we will state and prove the main results of the paper
2 Anisotropic variable exponent Sobolev spaces
We recall in what follows some definitions and basic properties of the generalized Lebesgue–Sobolev spaces L p(x)(Ω)
and W1,p(x)(Ω)whereΩis an open subset of RN In that context, we refer to the book of Musielak [7], the papers of Kováčik and Rákosník [6] and Fan et al [2,3] Set
C+(Ω) := {h; h∈C(Ω),h(x) >1 for all x∈Ω}
For any h∈C+(Ω)we define
h+=sup
x∈ Ω
h(x) and h−= inf
x∈ Ωh(x).
For any p(x) ∈C+(Ω), we define the variable exponent Lebesgue space
L p(x)(Ω) =
u:a measurable real-valued function such that
Ω
|u(x)|p(x)dx< ∞
We recall the following so-called Luxemburg norm on this space defined by the formula
|u|p(x)=inf
µ >0;
Ω
u(x) µ
p(x)
dx≤1
Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are Banach spaces, the Hölder inequality holds, they are reflexive if and only if 1 <p− ≤p+ < ∞and continuous functions are dense if p+ < ∞ The inclusion between Lebesgue spaces also generalizes naturally: if 0 < |Ω| < ∞and p1,p2are variable exponents so that
p1(x) ≤ p2(x)a.e x ∈Ωthen there exists the continuous embedding L p2 (x)(Ω) ↩→ L p1 (x)(Ω) We denote by L p′(x)(Ω)the
conjugate space of L p(x)(Ω), where p1(x)+p′1(x) =1 For any u∈L p(x)(Ω)andv ∈L p′(x)(Ω)the Hölder inequality
Ω
uvdx
≤
1
(p′)−
|u|p(x)| v|p′(x)
holds true
An important role in manipulating the generalized Lebesgue–Sobolev spaces is played by the modular of the L p(x)(Ω) space, which is the mappingρp(x):L p(x)(Ω) →R defined by
ρp(x)(u) =
Ω
|u|p(x)dx.
If u∈L p(x)(Ω)and p+< ∞then the following relations hold
provided|u|p(x)>1 while
provided|u|p(x)<1 and
Next, we define the space W1,p(x)
0 (Ω)as the closure of C∞
0 (Ω)under the norm
∥u∥p(x)= |∇u|p(x).
We point out that the above norm is equivalent with the following norm
∥u∥p(x)=
N
| ∂x i u|p(x),
Trang 4provided that p(x) ≥ 2 for all x ∈ Ω The space
W1,p(x)
0 (Ω), ∥.∥p(x)
is a separable and Banach space We note that if
s∈C+(Ω)and s(x) <p∗(x)for allΩthen the embedding
W1,p(x)
0 (Ω) ↩→L s(x)(Ω)
is compact and continuous, where p∗(x) = Np(x)
N−p(x)if p(x) <N or p∗(x) = ∞if p(x) >N.
We introduce a natural generalization of the variable exponent Sobolev space W1 ,p(x)(Ω)that will enable us to study problem(1.1)with sufficient accuracy Define− →p : Ω →RNthe vectorial function− →p = (p1,p2, ,p N) We introduce
the anisotropic variable exponent Sobolev space, W1, − →
p(x)
0 (Ω), as the closure of C∞
0 (Ω)with respect to the norm
∥u∥− →
p(x)=
N
i= 1
| ∂x i u|p i(x).
Then W1, − →
p(x)
0 (Ω)is a reflexive and separable Banach space, see [11–13] In the case when p iare all constant functions the
resulting anisotropic space is denoted by W1, − →
p
0 (Ω), where− →p is the constant vector(p1,p2, ,p N) The theory of such spaces has been developed in [9,10] Let us introduce− →P+, − →P−∈RN and P+,P+,P−,P−∈R+as
−
→
P+= (p+1,p+2, ,p+N), − →P−= (p−1,p−2, ,p−N)
and
P++=max{p+1,p+2, ,p+N} , P−+=max{p−1,p−2, ,p−N} ,
P−=min{p+1,p+2, ,p+N} , P−=min{p−1,p−2, ,p−N}
Throughout this paper we assume that
N
i= 1
1
and define P∗
−∈R and P−,∞∈R+by
N
i= 1
1
p−i −1
, P− ,∞=max{P−+,P−∗}
We recall that if s∈C+(Ω)satisfies 1<s(x) <P− ,∞for all x∈Ωthen the embedding W1, − →
p(x)
0 (Ω) ↩→L s(x)(Ω)is compact, see for example [13, Theorem 1]
3 Main results
In this section, we state and prove the main results of this paper We will use the letter C ito denote positive constants Let us introduce the following hypotheses:
(f0) f :Ω×R→R is a Carathéodory function and satisfies the subcritical growth condition
|f(x,t)| ≤C(1+ |t|q(x)− 1), ∀(x,t) ∈Ω×R,
where P++<q−≤q+<P−∗ and C is a positive constant;
(f1) f(x,t) =o(|t|P+−1), t →0, uniformly a.e x∈Ω;
(f2) lim|t|→+∞F(x,t)
|t|P
+ = +∞uniformly a.e x∈Ω, i.e f is P+-superlinear at infinity;
(f3) There exists a constant C∗>0 such that
G(x,t) ≤G(x,s) +C∗
for any x∈Ω, 0<t <s or s<t <0, where G(x,t) :=tf(x,t) −P++F(x,t)and F(x,t) := t
0f(x,s)ds;
(f4) f(x, −t) = −f(x,t)for all(x,t) ∈Ω×R.
It should be noticed that the condition(f3)is a consequence of the following condition, which was firstly introduced by O.H Miyagaki et al [21] and developed by G Li et al [22] and C Ji [23]:
(f′
3) There exists t0>0 such that f(x,t)
P+ 2 is nondecreasing in t≥t0and nonincreasing in t≤ −t0for any x∈Ω
Trang 5The readers may consult the proof and comments on this assertion in the papers [21–23] and the references cited there By some simple computations, we can show that the function
f(x,t) = |t|P+−2t log(1+ |t| ), t ∈R
satisfies our conditions(f0)–(f4)but it does not satisfy the (A–R) type condition(1.3)
Definition 3.1 A function u∈W1, − →
p(x)
0 (Ω)is said to be a weak solution of problem(1.1)if
N
i= 1
Ω
| ∂x i u|p i(x)− 2∂x i u· ∂x ivdx−
Ω
f(x,u)vdx=0
for allv ∈W1, − →
p(x)
Our main results in this paper are given by the following two theorems
Theorem 3.2 Assume that the conditions(f0)–(f3)are satisfied Then problem(1.1)has a non-trivial weak solution.
Theorem 3.3 Assume that the conditions(f0),(f2)–(f4)are satisfied Then problem(1.1)has infinitely many weak solutions{u k}
satisfying
N
i= 1
Ω
1
p i(x) | ∂x i u k|p i(x)dx−
Ω
F(x,u k)dx→ +∞ , k→ ∞
to problem(1.1)considered in anisotropic variable exponent Sobolev spaces (note that in this paper, we do not use the parameterλas in [21–23]), while ourTheorem 3.3seems to be new even with the well-known p-Laplace operator∆p u In
order to prove the main theorems, we recall some useful concepts and results
Definition 3.4 Let(X, ∥.∥)be a real Banach space, J∈C1(X,R) We say that J satisfies the(C c)condition if any sequence {u m} ⊂X such that J(u m) →c and∥J′(u m)∥(1+ ∥u m∥ ) →0 as m→ ∞has a convergent subsequence
Proposition 3.5 (See [ 24 ]) Let(X, ∥.∥)be a real Banach space, J∈C1(X,R)satisfies the(C c)condition for any c>0, J(0) =0
and the following conditions hold:
(i) There exists a functionφ ∈X such that∥ φ∥ > ρand J(φ) <0;
(ii) There exist two positive constantsρand R such that J(u) ≥R for any u∈X with∥u∥ = ρ.
Then the functional J has a critical value c ≥R, i.e there exists u∈X such that J′(u) =0 and J(u) =c.
In order to proveTheorem 3.3we will use the following fountain theorem, see [25] for details Let(X, ∥.∥)be a real
reflexive Banach space presenting by X = ⊕j∈NX jwith dim(X j) < +∞for any j∈N For each k∈N, we set Y k= ⊕k
j= 0X j
and Z k= ⊕∞j=k X j
Proposition 3.6 (See [ 25 ]) Let(X, ∥.∥)be a real reflexive Banach space, J∈C1(X,R)satisfies the(C c)condition for any c>0
and J is even If for each sufficiently large k∈N, there existρk>r k>0 such that the following conditions hold:
(i) a k:=inf{u∈Z k: ∥u∥=r k}J(u) → +∞as k→ ∞;
(ii) b k:=max{u∈Y k: ∥u∥= ρk}J(u) ≤0.
Then the functional J has an unbounded sequence of critical values, i.e there exists a sequence{u k} ⊂X such that J′(u k) =0 and
J(u k) → +∞as k→ +∞.
In the rest of this paper we will use the letter X to denote the anisotropic variable exponent Sobolev space W1, − →
p
0 (Ω) Let
us define the energy functional J :X→R by the formula
J(u) = N
i= 1
Ω
1
p i(x) | ∂x i u|p i(x)dx−
Ω
By the hypothesis(f0)and the continuous embeddings, some standard arguments assure that the functional J is well-defined
on X and J∈C1(X)with the derivative given by
J′(u)(v) =
N
i= 1
Ω
| ∂x i u|p i(x)− 2∂x i u· ∂x ivdx−
Ω
f(x,u)vdx
for all u, v ∈X Thus, weak solutions of problem(1.1)are exactly the critical points of the functional J.
Trang 6Lemma 3.7 Assume that the conditions(f0)–(f2)are satisfied Then we have the following assertions:
(i) There existsφ ∈X ,φ >0 such that J(tφ) → −∞as t→ +∞;
(ii) There existρ >0 and R>0 such that J(u) ≥R for any u∈X with∥u∥ = ρ.
Proof (i) From(f2), it follows that for any M >0 there exists a constant C M=C(M) >0 depending on M, such that
Takeφ ∈X withφ >0, from(3.2)we get
J(tφ) = N
i= 1
Ω
| ∂x i tφ|p i(x)
p i(x) dx−
Ω
F(x,tφ)dx
≤ t P+
P−
N
i= 1
Ω
| ∂x iφ|p i(x)dx−Mt P+
Ω
| φ|P+dx+C M|Ω|
≤t P+
1
P−
N
i= 1
Ω
| ∂x iφ|p i(x)dx−M
Ω
| φ|P+dx
where t>1 and|Ω|denotes the Lebesgue measure ofΩ From(3.3), if M is large enough such that
1
P−
N
i= 1
Ω
| ∂x iφ|p i(x)dx−M
Ω
| φ|P+dx<0,
then we have
lim
t→+∞J(tφ) = −∞,
which ends the proof of (i)
(ii) Since the embeddings X↩→L P+(Ω)and X↩→L q(x)(Ω)are continuous, there exist constants C1,C2>0 such that
∥u∥
L P
+
( Ω )≤C1∥u∥−→p(x), ∥u∥L q(x) ( Ω )≤C2∥u∥− →
Let 0< ϵC P
+
2P + N+ P
+ 1, where C1is given by(3.4) From(f0)and(f1), we have
Let u∈X with∥u∥− →p(x)<1 sufficiently small For such an element u we get| ∂x i u|p i(x)<1 for all i=1,2, ,N Using(2.1)
and some simple computations, we obtain
N
i= 1
Ω
| ∂x i u|p i(x)dx≥ N
i= 1
| ∂x i u|p
+
i
p i(x)
≥
N
i= 1
| ∂x i u|P +
p i(x)
≥N
N
i= 1
| ∂x i u|p i(x)
N
P+
=
∥u∥P +
−
→
p(x)
Trang 7From(3.4)–(3.6)we have
Jλ(u) =
N
i= 1
Ω
| ∂x i u|p i(x)
p i(x) dx−
Ω
F(x,u)dx
≥
∥u∥P
+
−
→
p(x)
P++N P+−1
− ϵ
Ω
|u|P+dx−C(ϵ)
Ω
|u|q(x)dx
P+N P+−1
∥u∥P +
−
→p(x)− ϵC P
+
1 ∥u∥P +
−
→p(x)−C(ϵ)C2q−∥u∥q−→−
p(x)
≥
1
2P++N P+−1
−C(ϵ)C2q−∥u∥q
− −P+
−
→
p(x)
∥u∥P +
−
→
where C2>0 is given by(3.4) From(3.7)and the fact that q−>P++, we can choose R>0 andρ >0 such that J(u) ≥R>0
for all u∈X with∥u∥− →
p(x)= ρ The proof ofLemma 3.7is complete
Lemma 3.8 Assume that the conditions(f0),(f2)–(f3)are satisfied Then the functional J satisfies the(C c)condition for any c>0.
Proof Let{u m} ⊂X be a(C c)sequence of the functional J, that is,
J(u m) →c>0, ∥J′(u m)∥∗(1+ ∥u m∥− →
p(x)) →0 as m→ ∞ , which shows that
where o(1) →0 as m→ ∞
We will prove that the sequence {u m} is bounded in X Indeed, if{u m} is unbounded in X , we may assume that
∥u m∥− →
p(x) → ∞as m → ∞ We define the sequence{ wm}bywm = u m
∥u m∥ − →p(x), m = 1,2, It is clear that{ wm} ⊂ X
and∥ wm∥− →
p(x) = 1 for any m Therefore, up to a subsequence, still denoted by{ wm}, we have{ wm}converges weakly to
LetΩ̸=:= {x∈Ω : w(x) ̸=0} If x∈Ω̸=then it follows from(3.9)that|u m(x)| = |wm(x)|∥u m∥− →
p(x)→ +∞as m→ ∞.
Moreover, from(f2), we have
lim
m→∞
F(x,u m(x))
Using the condition(f2), there exists t0>0 such that
F(x,t)
for all x∈Ωand|t| >t0>0 Since F(x,t)is continuous onΩ× [−t0,t0], there exists a positive constant C3such that
for all(x,t) ∈Ω× [−t0,t0] From(3.13)and(3.14)there exists C4∈R such that
for all(x,t) ∈Ω×R From(3.15), for all x∈Ωand m, we have
F(x,u m(x)) −C4
∥u m∥P
+
−
→ ( )
≥0
Trang 8F(x,u m(x))
|u m(x)|P+ | wm(x)|P+− C4
∥u m∥P +
−
→
p(x)
For each i∈ {1,2, ,N}and m∈N, we define
αi,m=
P++ if| ∂x i u m|p i(x)<1,
P−− if| ∂x i u m|p i(x)>1.
Using(2.1),(2.2)and some simple computations, we infer that for any m,
N
i= 1
Ω
| ∂x i u m|p i(x)dx≥ N
i= 1
| ∂x i u m|αi,m
p i(x)
≥
N
i= 1
| ∂x i u m|P
−
p i(x)−
{i: αi,m=P+}
| ∂x i u m|P
−
p i(x)− | ∂x i u m|P
+
p i(x)
≥ N
N
i= 1
| ∂x i u m|p i(x)
N
P−
−N
=
∥u m∥P
−
−
→
p(x)
N P−−1
c= J(u m) +o(1)
=
N
i= 1
Ω
| ∂x i u m|p i(x)
p i(x) dx−
Ω
F(x,u m)dx+o(1)
≥
∥u m∥P
−
−
→
p(x)
P+N P−− 1
P+−
Ω
F(x,u m)dx+o(1) or
Ω
F(x,u m)dx≥ 1
P+N P−− 1
∥u m∥P
−
−
→
p(x)−c−
N
Similarly, for each i∈ {1,2, ,N}and m∈N, we define
βi,m=
P− if| ∂x i u m|p i(x)<1,
P++ if| ∂x i u m|p i(x)>1.
Using(2.1),(2.2)and some simple computations, we infer that for any m,
N
i= 1
Ω
| ∂x i u m|p i(x)dx≤ N
i= 1
| ∂x i u m|βi,m
p i(x)
≤
N
i= 1
| ∂x i u m|P +
p i(x)−
{i: βi,m=P−}
| ∂x i u m|P +
p i(x)− | ∂x i u m|P
−
p i(x)
+N
≤
N
i= 1
| ∂x i u m|p i(x)
P+
+2N
= ∥u m∥P
+
−
Trang 9Thus, we have from(3.8)and(3.19)that
c =J(u m) +o(1)
=
N
i= 1
Ω
| ∂x i u m|p i(x)
p i(x) dx−
Ω
F(x,u m)dx+o(1)
P−−∥u m∥
P+
−
→
p(x)+
2N
P−− −
Ω
F(x,u m)dx+o(1)
or by(3.18),
∥u m∥P
+
−
→p(x)≥P
−
−
Ω
F(x,u m)dx+cP−−−2N−o(1) >0 for m large enough. (3.20)
We claim that|Ω̸=| =0 In fact, if|Ω̸=| ̸=0, then by(3.12),(3.16),(3.20)and the Fatou lemma, we have
+ ∞ =
Ω̸=
lim
m→∞
F(x,u m(x))
|u m(x)|P+ | wm(x)|P+dx−
Ω̸=
lim
m→∞
C4
∥u m∥P +
−
→
p(x)
dx
=
Ω̸=
lim
m→∞
F(x,u m(x))
|u m(x)|P+ | wm(x)|P+− C4
∥u m∥P +
−
→
p(x)
dx
≤lim inf
m→∞
Ω̸=
F(x,u m(x))
|u m(x)|P+ | wm(x)|P+− C4
∥u m∥P+
dx
≤lim inf
m→∞
Ω
F(x,u m(x))
|u m(x)|P+ | wm(x)|P+− C4
∥u m∥P +
−
→
p(x)
dx
=lim inf
m→∞
Ω
F(x,u m(x))
∥u m∥P +
−
→
p(x)
dx−lim sup
m→∞
Ω
C4
∥u m∥P +
−
→
p(x)
dx
=lim inf
m→∞
Ω
F(x,u m(x))
∥u m∥P +
−
→p(x)
dx
≤lim inf
m→∞
ΩF(x,u m(x))dx
P−
P−−,
which is a contradiction This shows that|Ω̸=| =0 and thusw(x) =0 a.e inΩ
Since J(tu m)is continuous in t∈ [0,1], for each m there exists t m∈ [0,1], m=1,2, ,such that
J(t m u m) := max
It is clear that t m > 0 and J(t m u m) ≥ c > 0 = J(0) = J(0.u m) If t m < 1 then dt d J(tu m)|t=t m = 0 which gives
J′(t m u m)(t m u m) =0 If t m=1, then J′(u m)(u m) =o(1) So we always have
Let{R k}be a positive sequence of real numbers such that R k>1 for any k and lim k→∞R k= +∞ Then∥R kwm∥− →p(x)=R k>1
for any k and m Fix k, sincewm →0 strongly in the spaces L q(x)(Ω)andwm(x) → 0 a.e x ∈ Ωas m → ∞, using the condition(f0)and the Lebesgue dominated convergence theorem we deduce that
lim
m→∞
Ω
Trang 10Since∥u m∥− →
p(x)→ ∞as m→ ∞, we also have∥u m∥− →
p(x)>R kor 0< R k
∥u m∥ − →p(x) <1 for m large enough Hence, using(3.17)
J(t m u m) ≥J
R k
∥u m∥− →
p(x)
u m
= J(R kwm)
=
N
i= 1
Ω
| ∂x i R kwm|p i(x)
p i(x) dx−
Ω
F(x,R kwm)dx
P+.
∥R kwm∥P
−
−
→p(x)
N P−−1
−N
−
Ω
F(x,R kwm)dx
−
k
P++N P−−1
P++−
Ω
F(x,R kwm)dx
−
k
2P+N P−−1
for any m large enough From(3.25), letting m,k→ ∞we have
lim
On the other hand, using the condition(f3)and(3.8), for all m large enough, we have
J(t m u m) = J(t m u m) − 1
P++J
′(t m u m)(t m u m) +o(1)
=
N
i= 1
Ω
| ∂x i t m u m|p i(x)
p i(x) dx−
Ω
F(x,t m u m)dx
− 1
P++
N
i= 1
Ω
| ∂x i t m u m|p i(x)dx+ 1
P++
Ω
f(x,t m u m)t m u m dx+o(1)
=
N
i= 1
Ω
1
p i(x) −
1
P++
| ∂x i t m u m|p i(x)dx+ 1
P++
Ω
G(x,t m u m)dx+o(1)
≤
N
i= 1
Ω
1
p i(x) −
1
P+
| ∂x i u m|p i(x)dx+ 1
P+
Ω
G(x,u m) +C∗
dx+o(1)
=
N
i= 1
Ω
| ∂x i u m|p i(x)
p i(x) dx−
Ω
F(x,u m)dx
− 1
P++
N
i= 1
Ω
| ∂x i u m|p i(x)dx−
Ω
f(x,u m)u m dx
+ C∗
P++|Ω| +o(1)
= J(u m) − 1
P+J
′(u m)(u m) + C∗
P+|Ω| +o(1)
→c+ C∗
Now, since the Banach space X is reflexive, there exists u∈X such that passing to a subsequence, still denoted by{u m},
it converges weakly to u in X and converges strongly to u in the space L q(x)(Ω) Using the condition(f)and the Hölder