Journal of Applied Mathematics and Physics, 2015, 3, 1270-1281 Published Online October 2015 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2015.310156
Existence and Multiplicity of Solutions for Quasilinear p(x)-Laplacian Equations in N Honghong Qi, Gao Jia College of Science, University of Shanghai for Science and Technology, Shanghai, China Email:
[email protected],
[email protected] Received 10 September 2015; accepted 25 October 2015; published 28 October 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/
Abstract We establish some results on the existence of multiple nontrivial solutions for a class of p(x)-Laplacian elliptic equations without assumptions that the domain is bounded. The main tools used in the proof are the variable exponent theory of generalized Lebesgue-Sobolev spaces, variational methods and a variant of the Mountain Pass Lemma.
Keywords p(x)-Laplacian Operator, Generalized Lebesgue-Sobolev Spaces, Variational Method, Multiple Solutions
1. Introduction The study of differential and partial differential equations involving variable exponent conditions is a new and interesting topic. The interest in studying such problem was stimulated by their applications in elastic mechanics and fluid dynamics. These physical problems were facilitated by the development of Lebesgue and Sobolev spaces with variable exponent. The existence and multiplicity of solutions of p ( x ) -Laplacian problems have been studied by several authors (see for example [1] [2], and the references therein). In [3], A. R. EL Amrouss and F. Kissi proved the existence of multiple solutions of the following problem
(
)
( x )−2 −div ∇u p= ∇u = u 0,
f ( x, u ) , in Ω,
(1)
on ∂Ω.
How to cite this paper: Qi, H.H. and Jia, G. (2015) Existence and Multiplicity of Solutions for Quasilinear p(x)-Laplacian EquN ations in . Journal of Applied Mathematics and Physics, 3, 1270-1281. http://dx.doi.org/10.4236/jamp.2015.310156
H. H. Qi, G. Jia
Also Xiaoyan Lin and X. H. Tang in [4] studied the following quasilinear elliptic equation
(
)
−div ∇u p − 2 ∇u + v ( x ) u u ∈ W 1, p N ,
p−2
u= f ( x, u ) , in N ,
( )
(2)
and they proved the multiplicity of solutions for problem (2) by using the cohomological linking method for cones and a new direct sum decomposition of W 1, p N . In this paper, we consider the following problem
( )
p( x)−2 u = f ( x, u ) , in N , −∆ p ( x ) u + b ( x ) u 1, p ( x ) N , u ∈ W
(3)
( )
(
where ∆ p( x )u = div ∇u function with
p( x )−2
∇u
)
is the p ( x ) -Laplacian operator; p ( x ) : N → is a Lipschitz continuous
= 1 < p − infN p ( x ) ≤ p (= x ) ≤ p + sup p ( x ) < ∞. x∈
x∈ N
b ( x ) is a given continuous function which satisfies (B0)
(
)
b − > 0, m b −1 ( 0, T ] < +∞, for all T ∈ + , here m is the Lebesgue measure on N. f : N × → is a Carathéodory function satisfying the subcritical growth condition (F0)
(
q ( x ) −1
f ( x, t ) ≤ c 1 + t
( )
) , ∀t ∈ , a.e. x ∈
N
,
infN q ( x ) , 1 < q ( x ) < p∗ ( x ) , ∀x ∈ , and for some c > 0 , where q ( x ) ∈ C N , p + < q − = x∈
Np ( x ) , p ( x) = N − p ( x) ∞,
p ( x) < N,
∗
Define the subspace
{u ∈W
E=
1, p ( x )
( ) ∫ N
N
( ∇u
p ( x) ≥ N.
p( x)
+ b ( x) u
p( x)
) dx < +∞}
and the functional Φ : E → ,
= Φ (u )
∫
N
(
1 ∇u p ( x)
p( x)
+ b ( x) u
p( x)
) dx − ∫
N
F ( x, u ) dx, ∀u ∈ E ,
where F ( x, u ) = ∫0 f ( x, t ) dt . Clearly, in order to determine the weak solutions of problem (3), we need to find the critical points of functional Φ. It is well known that under (B0) and (F0), Φ is well defined and is a C1 functional. Moreover, u
Φ′ ( u ) ,= v
∫
N
( ∇u
p( x)−2
∇u ∇v + b ( x ) u
p( x)−2
)
uv dx − ∫
N
f ( x, u ) vdx,
for all u , v ∈ E . If f ( x, 0 ) = 0 for a.e. x ∈ N , the constant function u = 0 is a trivial solution of problem (3). In the following, the key point is to prove the existence of nontrivial solutions for problem (3). Set
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H. H. Qi, G. Jia
∫
= λ∗
N
inf
(
1 ∇u p ( x)
u∈E \{0}
∫
N
p( x )
+ b ( x) u
1 u p ( x)
p( x )
p( x )
) dx
> 0.
(4)
dx
This paper is to show the existence of nontrivial solutions of problem (3) under the following conditions. (F1) 2 p− p− lim F ( x, t ) − λ t = −∞, uniformly for a.e. x ∈ N , ∗ t →∞ + 3 p where λ* as given in (4). (F2) There exist µ ∈ 1, p − and γ > 0 , such that
( ) ( )
)
0 < µ F ( x, t ) ≤ tf ( x, t ) , for a.e. x ∈ N , 0 < t ≤ γ .
(F3) There exist θ > p + and K > 0 such that
t ≥ K ⇒ 0 < θ F ( x, t ) ≤ tf ( x, t ) , for a.e. x ∈ , ∀t ∈ . N
(
(F4) f ( x, t ) = o t
p + −1
)
as t → 0 and uniformly for x ∈ N , with q − > p + . Here q ( x ) is given in the
condition (F0). We have the following results. Theorem 1.1. If b ( x ) satisfies (B0), f ( x ) satisfies (F0), (F1) and (F2), then problem (3) possesses at least one nontrivial solution. Theorem 1.2. Assume b ( x ) satisfies (B0), f ( x ) satisfies (F0), (F3) and (F4), with f ( x, 0 ) = 0 for a.e. x ∈ N , then problem (3) has at least two nontrivial solutions, in which one is non-negative and another is non-positive. This paper is divided into three sections. In the second section, we state some basic preliminary results and give some lemmas which will be used to prove the main results. The proofs of Theorem 1.1 and Theorem 1.2 are presented in the third section.
2. Preliminaries In this section, we recall some results on variable exponent Sobolev space W ( p x the variable exponent Lebesgue space L ( ) N , we refer to [5]-[8]. N ∞ − Let p ( x ) ∈ L , p > 1 . Define the variable exponent Lebesgue space:
1, p x )
( )
( ) ) L(= ( ) {u : → u is a measurable function and ∫ u ∈ L ( ) ( ) , we define the following norm p x
N
N
N
For
p x
u
( )
p( x)
N
and basic properties of
}
dx < +∞ .
N
u
p( x)
p( x)
u ( x) = inf µ > 0 ∫ N µ
dx ≤ 1 .
Define the variable exponent Sobolev space:
W
1, p ( x )
( =) {u : N
N
p( x)
→ u∈L
( ) and ∇u ∈ L ( ) ( )} , p x
N
N
which is endowed with the norm p( x)
It can be proved that the spaces L See [9] for the details. Proposition 2.1. [10] [11] Let
u 1,= u p( x)
( ) N
p( x)
and W
1272
+ ∇u
1, p ( x )
p( x)
( ) N
. are separable and reflexive Banach spaces.
= J (u )
∫
N
u
p( x )
p( x )
dx, u ∈ L
H. H. Qi, G. Jia
( ). N
Then we have 1) For u ≠ 0 , u 2) u 3) u
p( x ) p( x )
4) un
>1⇒ u
p( x )
p− p( x )
u 1; µ ⇔ J = = µ
≤ J (u ) ≤ u
p+ p( x )
, u
p( x )
1( =1, < 1) ⇔ J ( u ) > 1( =1, < 1) ;
p( x )
→ 0 ⇔ J ( un ) → 0 , un
( )
For h ( x ) ∈ L ∞
N
p( x )
→ ∞ ⇔ J ( un ) → ∞.
with h > 1 , let h∗ ( x ) : N → satisfy −
1 1 + = 1, a.e. x ∈ N . h ( x ) h∗ ( x ) We have the following generalized Hölder type inequality. ∗ h x Proposition 2.2. [9] [12] For any u ∈ L ( ) N and v ∈ Lh ( x ) N , we have
( )
( )
1 1 ∫ N uvdx ≤ h− + h∗− u h( x ) v h∗ ( x ) .
We consider the case that b ( x ) satisfies (B0). Define the norm
u => inf µ 0
∇u ∫ N µ
p( x)
+ b ( x)
u
p( x)
µ
( )
dx ≤ 1 .
Then ( E , ⋅ ) is continuously embedding into W ( ) N as a closed subspace. Therefore, a separable and reflexive Banach space. Similar to the Proposition 2.1, we have 1, p x Proposition 2.3. [13] The functional J1 : W ( ) N → defined by 1, p x
(
( )
J1 ( u ) = ∫ N ∇u
p( x )
+ b ( x) u
p( x )
( E, ⋅ )
is also
) dx
has the following properties: u J1 = 1; µ 1) u ≠ 0, u =⇔ µ p− p+ p+ p− 2) u > 1 ⇒ u ≤ J1 ( u ) ≤ u , u < 1 ⇒ u ≤ J1 ( u ) ≤ u ; 3) un → 0 ⇔ J1 ( un ) → 0. Lemma 2.4. [13] If b ( x ) satisfies (B0), then p x L ( ) N ; 1) we have a compact embedding E 2) for any measurable function q : N → with p ( x ) ≤ q ( x ) p∗ ( x ) , we have a compact embedding
( )
E
L(
q x)
( ) . Here N
u v means that infN ( v ( x ) − u ( x ) ) > 0 . x∈
Now, we consider the eigenvalues of the p(x)-Laplacian problem −∆ u + b ( x ) u p( x )−2 u = λ u p( x ) 1, p ( x ) N . u ∈ W
( )
p( x )−2
u , in N ,
(5)
For any u ∈ E , define G, H : E → by
G= (u )
∫
N
(
1 ∇u p ( x)
p( x )
+ b ( x) u
For all t > 0 , set
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p( x )
(u ) ) dx, H=
∫
N
1 u p ( x)
p( x )
dx.
H. H. Qi, G. Jia
St = H −1 ( t ) = t} , {u ∈ E : H ( u ) =
then St is a C1 submanifold of E since t is a regular value of H. Put
− I , γ ( I ) ≥ n} , { I ⊂ St : I = ∑= t ,n
where γ ( I ) is the genus of I. Define c( n= inf sup G ( u ) , = n 1, 2, ⋅⋅⋅. ,t )
We denote by
{(u
( n ,t ) , λ( n ,t )
)}
I ∈ ∑ u∈I t ,n
the eigenpair sequences of problem (5) such that
(
(
)
)
H ±u( n,t ) = t , G ±u( n,t ) = c( n,t ) ,
= λ( n ,t )
∫
N
p( x)
∇u ( n ,t )
∫
N
+ b ( x ) u( n ,t )
u( n ,t )
Define
µ∗ = inf
∫
N
E \{0}
p( x)
( ∇u
p( x)
dx → ∞, as n → ∞.
dx
p( x )
∫
u
N
µ∗ = inf
E \{0}
+ b ( x) u p( x )
p( x )
) dx ,
dx
G (u ) , H (u )
λ= inf Λ , where Λ= {λ ∈ : λ is an eigenvalue of ( 5 )}. ∗ Lemma 2.5. For all t > 0 , let u(1,t ) be an eigenfunction associated with λ(1,t ) of the problem (5). Then,
( )
G u(1,= c= t) (1,t ) inf {G ( u ) : u ∈ St } .
Proof. Let zt inf {G ( u ) : u ∈ St } . From the definition of c(1,t ) , it is easy to see that zt ≤ c(1,t ) . = On the other hand, since the functional G : E → is coercive and weakly lower semi-continuous and St is zt . Letting I = {±u0 } , then γ ( I ) = 1 and weakly closed subset of E, there exists u0 ∈ St such that G ( ±u0 ) = c(1,t ) ≤ zt . Thus the lemma follows. Lemma 2.6. 2
2
p− p+ µ λ ≤ ≤ + ∗ − µ∗ . ∗ p p Proof. From Lemma 2.5, we have
( ) ( )
G u(1,t ) G (u ) = inf u ∈ St . H u(1,t ) H (u )
Since
( )=∫ H ( u( ) ) G u(1,t )
N
1, t
−
p ≤ + p
p( x) 1 + b ( x ) u(1,t ) ∇u(1,t ) p ( x) p( x) 1 ∫ N p ( x ) u(1,t ) dx
∫
N
∇u (1,t )
∫
1274
p( x)
N
+ b ( x ) u(1,t )
u(1,t )
p( x)
dx
p( x)
p( x)
dx
dx ,
so we have
H. H. Qi, G. Jia
G (u ) p− u ∈ St . Then, λ ≤ inf + (1, t ) p H (u )
p− G (u ) u ∈ St , for all t > 0. λ ≤ inf + ∗ H u p ( )
Thus we get
p− p+ λ ≤ µ∗ and λ∗ ≤ − µ∗ . + ∗ p p
Similarly, if u( n ,t ) is the eigenfunction associated with λ( n ,t ) , we get λ( n,t ) ≥ ly, we obtain
p− p− λ∗ ≥ + µ∗ . Finalµ + ∗ and p p
p− p+ µ∗ ≤ λ∗ ≤ − µ∗ . + p p p− p+ µ ≤ µ ≤ µ∗ . Thus the lemma follows. ∗ ∗ p+ p−
On the other hand, it is easy to see that Now, we consider the truncated problem
−∆ u + b ( x ) u p( x )−2 u = f ( x, u ) , in N , ± p( x ) 1, p ( x ) N , u ∈ W
(M± )
( )
where f ( x, t ) , if ± t ≥ 0, f ± ( x, t ) = otherwise. 0,
We denote by u ( ) = max ( u , 0 ) and = u ( ) max ( −u , 0 ) the positive and negative parts of u. Lemma 2.7. + − 1) If u ∈ E then u ( ) , u ( ) ∈ E and −
+
∇u , u > 0, 0, u ≥ 0, + − = ∇u ( ) = ∇u ( ) 0, u ≤ 0, ∇u , u < 0. 2) The mappings u → u ( ± ) are continuous on E. Lemma 2.8. All solutions of ( M − ) (resp. M + ) are non-positive (resp. non-negative) solutions of problem (3). Proof. Define Φ ± ( u ) : E → ,
Φ ±= (u )
∫
=
∫
N
N
( 1 ( ∇u p ( x) 1 ∇u p ( x)
p( x)
p( x)
+ b ( x) u
p( x)
+ b ( x) u
p( x)
) dx − ∫ F ( x, u ) dx ) dx − ∫ F ( x, u ) dx, N
±
(±)
N
where F± ( x, s ) = ∫0 f ± ( x, t ) dt. From Lemma 2.7 and (F0), Φ ± is well defined on E, weakly lower semi-continuous and C1-functionals. Let u be a solution of ( M − ) . Taking v = u ( + ) in s
Φ ′− ( u = )v
∫
N
( ∇u
p( x)−2
∇u ∇v + b ( x ) u
p( x)−2
)
uv − f − ( x, u ) v d= x 0,
we have
( ) ) =∫
J1 u (
+
N
(+) ∇u
p( x)
+ b ( x ) u(
+)
p( x)
0. dx =
By virtue of Proposition 2.3, we have u ( + ) = 0 , so u ( + ) = 0 and u = u ( − ) , a.e. x ∈ N , then u is also a criti-
1275
H. H. Qi, G. Jia
Φ− (u ) . cal point of the functional Φ with critical value Φ ( u ) = Similarly, the nontrivial critical points of the functional Φ + are non-negative solutions of problem
(M+ ) .
3. Proof of Main Results 3.1. Proof of Theorem 1.1 To derive the Theorem 1.1, we need the following results. Proposition 3.1. Φ is coercive on E. Proof. Put
(p ) L= ( x, t ) F ( x, t ) − (p )
− 2 +
3
p−
λ∗ t
.
From (F1) we have, for any R > 0 , there is M R > 0 such that L ( x, t ) ≤ − R, ∀ t ≥ M R , a.e. x ∈ N . By contradiction, let A ∈ and
Putting vn =
{un } ⊂ E
such that
un → ∞ and Φ ( un ) ≤ A.
(6)
un , one has vn = 1 . For a subsequence, we may assume that for some v0 ∈ E , we have un
( )
vn v0 weakly in E and vn → v0 strongly in L ( ) N . Consequently, v0 ≡/ 0 . Let Ω= x ∈ N : v0 ≡/ 0 , via the result above we have Ω ≡/ 0 and p x
{
}
un → +∞, a.e. x ∈ Ω.
Set un ( x ) , un = 0,
x ∈ Ω, x ∈ N \ Ω,
then,
un → +∞, as n → ∞, a.e. x ∈ Ω. From (6), (F1) and Lemma 2.6, we deduce that 1 A= ≥ Φ ( un ) ∫ N ∇u n p ( x)
(
≥
1 p+
∫Ω ∇un
p( x )
p( x )
+ b ( x ) un
+ b ( x ) un
p( x )
2
p( x )
) dx − ∫
p− − + λ∗ un p
N
p( x )
F ( x, un ) dx
dx − ∫ L ( x, un ) dx Ω
≥ − ∫ΩL ( x, un ) dx → +∞.
This is a contradiction. Therefore, Φ is coercive on E. Proposition 3.2. Assume f ( x ) satisfies (F0) and (F2), then zero is local maximum for the functional Φ ( su ) , u ≠ 0 , s ∈ . Proof. From (F2), there is a constant c1 > 0 such that F ( x, t ) ≥ c1 t , for x ∈ N , 0 < t ≤ γ , 1 ≤ µ < p − . µ
(7)
From (F0) and t ≥ γ > 1 , there exists c2 > 0 such that F ( x, t ) ≤ c2 t
q( x)
, x ∈ N , t > γ .
(8)
By (7) and (8), we get F ( x, t ) ≥ c1 t − c2 t µ
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q( x)
, x ∈ N , t ∈ .
(9)
H. H. Qi, G. Jia
For u ∈ E , u ≠ 0, 0 < s < 1, we have −
Φ ( su ) ≤
sp p−
≤
sp p−
∫
N
∫
N
−
( ∇u
p( x )
( ∇u
p( x )
+ b ( x) u
p( x )
+ b ( x) u
p( x )
) dx − ∫ ( c s
µ
u − c2 s q u
) dx − c s
µ
+ c2 s q
1
N
µ
u
1
µ
Lµ
−
−
∫
N
u
q( x )
q( x )
) dx dx.
Since 1 ≤ µ < p − < q − , there is a= s0 s0 ( u ) > 0 such that
Φ ( su ) < 0, for all 0 < s < s0 < 1.
(10)
Thus the proposition follows. Proof of Theorem 1.1. From Proposition 3.1, we know Φ is coercive on E. Since Φ has a global minimizer 0 , then, in order to prove u0 ≠ 0 , we need to prove u0 on E, Φ is weakly lower semi-continuous and Φ ( 0 ) = Φ ( u0 ) < 0 . So we have the Theorem 1.1 following from Proposition 3.2.
3.2. Proof of Theorem 1.2 To find the properties of the p(x)-Laplacian operators, we need the following inequalities (see [10]). Lemma 3.3. For α and β in N, then there are the following inequalities
(
α
(α
n−2
n−2
α−β
α− β
n−2
n
)
(
n β (α − β ) 2 α + β
n−2
)
)
n
2−n 2
≥ ( n − 1) α − β , for 1 < n < 2; n
β (α − β ) ≥ 2 − n α − β ,
for n > 2.
n
on (1) Proposition 3.4. Assume (F0), and let {un } be a sequence such that un u in E and Φ ′− ( un ) v = for all v ∈ E as n → ∞ , then, for some subsequences, ∇un ( x ) → ∇u ( x ) , a.e. in N, as n → ∞ and Φ ′− ( u ) v = 0 for all v ∈ E . Proof. Let R > 0 and η ∈ C0∞ ( N ) such that 2 = η 0 if x ≥ 2 R, = η 1 if x ≤ R, 0 ≤ η ( x ) ≤ 1 for all x ∈ N and ∇η < . R Let us denote by { Pn } the following sequence
(
Pn = ∇un
p( x)−2
∇u n − ∇u
p( x)−2
)
∇u ( ∇u n − ∇u ) .
From Lemma 3.3, we have Pn ≥ 0 and
∫B (0)Pn dx ≤ ∫ R
N
∇u n
p( x)
− ∫ N ∇u
Recalling that un u in E, we get
∫
N
∇u
p( x)−2
and so,
∫B (0)Pn dx ≤ ∫ R
=
∫
N
∇u n
N
∇u n
⋅η dx − ∫ N ∇un
p( x)−2
p( x)−2
∇ u n ∇ u ⋅ η dx
∇u∇ ( un − u ) ⋅η dx.
η∇u∇ ( un − u ) dx → 0, as n → ∞, p( x)
η dx − ∫ N ∇un
p( x)−2
p( x)−2
η∇un ∇udx + on (1) (11)
η∇un ∇ ( un − u ) dx + on (1) .
On the other hand, by (11) and Φ ′− ( un ) η ( un − u ) = on (1) , we obtain
∫B (0)Pn dx ≤ on (1) − ∫ R
N
∇u n
− ∫ N b ( x ) un
p( x)−2
p( x)−2
u n ( u n − u ) η dx + ∫
N
≤ on (1) − ∫ N ∇un − ∫ N b ( x ) un
( u n − u ) ∇ u n ∇ η dx
p( x)−2
p( x)−2
(
f x, un(
−)
) (u
n
− u )η dx
( u n − u ) ∇ u n ∇ η dx
un ( un − u )η dx + ∫ N f ( x, un )( un − u )η dx.
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H. H. Qi, G. Jia
Thus, p ( x ) −1
∫B (0)Pn dx ≤ on (1) + c1 ∫supη ∇un R
p ( x ) −1
+ c1 ∫sup b ( x ) un
un − u dx
un − u dx + c2 ∫sup un
η
q ( x ) −1
η
un − u dx.
Combining Hölder’s inequality and Sobolev embedding, we deduce that
∫B (0)Pn dx → 0,
as → ∞.
(12)
R
Let us consider the sets
{
}
{
}
B1 =∈ x BR ( 0 ) 1 < p ( x ) < 2 and B2 =∈ x BR ( 0 ) p ( x ) ≥ 2 . From Lemma 3.3, we get
(
)
−
Pn ≥ p − 1
2
∇u n − ∇u
2
( ∇u n
+ ∇u )
+
Pn ≥ 2− p ∇un − ∇u
, if x ∈ B1 ,
2− p( x)
p( x )
(13)
, if x ∈ B2 .
(14)
Applying again Hölder’s inequality,
∫B
1
∇u n − ∇u
p( x)
dx ≤ C g n
2
p x L ( ) ( B1 )
hn
2 2− p( x )
L
( B1 )
,
(15)
where gn ( x ) =
∇u n − ∇u
( ∇u n
+ ∇u )
( ∇u n
+ ∇u )
p( x)
p ( x )( 2 − p ( x ) )
,
2
and hn ( x ) =
p ( x )( 2 − p ( x ) ) 2
.
Then,
∫B
1
hn
2 2− p( x)
dx =
∫B ( ∇un 1
+ ∇u )
p( x)
dx < +∞.
(16)
From (12) and (13), we have
∫B
1
gn
2 p( x)
dx ≤ C ∫ Pn dx → 0, as n → ∞.
(17)
B1
By (15)-(17), we obtain
∫B
∇u n − ∇u
∫B
∇u n − ∇u
1
p( x)
dx → 0, as n → ∞.
(18)
dx → 0, as n → ∞.
(19)
(12) and (14) imply that 2
p( x)
From (18) and (19), ∇un → ∇u a.e. in BR ( 0 ) . Because R is arbitrary, it follows that for some subsequence
∇un → ∇u a.e. in N . Combined with Lebesgue’s dominated convergence theorem, we get ∇u n
p( x)−2
∇u n ∇u
p( x)−2
1278
p( x) p x −1 ∇u in L ( ) N
( )
N
.
(20)
H. H. Qi, G. Jia
on (1) , we derive that Φ ′− ( u ) v = 0 for all v ∈ E . By (20) and Φ ′− ( un ) v = Proposition 3.5. Assume (F0), and let d ∈ and {un } be a (PS)d sequence in E for Φ − , then bounded in E. q( x) , ∀t ∈ , a.e. x ∈ N . It is clear that Proof. From (F0), we have F ( x, t ) ≤ c1 t Φ − ( un ) −
1
θ
Φ ′− = ( un ) un
∫ −
≥ ≥ ≥
N
1
θ
(
p( x)
1 ∇u n p ( x)
∫
N
( ∇u
p( x) n
(
p( x)
(−) n
N
) dx + θ1 ∫ f ( x, u ) u dx + b ( x) u ) dx − ∫ F ( x , u ) d x p( x)
+ b ( x ) un
1 1 − N ∇u n p + θ ∫
is
) dx − ∫ F ( x , u ) d x
p( x)
+ b ( x ) un
{un }
(−) n
N
p( x)
n
1 1 − − J1 ( un ) − c1 ∫ N un( ) + p θ 1 1 − J1 ( un ) − c1 ∫ N un p+ θ
q( x)
q( x)
N
n
(−) n
dx
dx.
Assume that un > 1 for some n ∈ N , then, by Proposition 2.3, Hölder’s inequality and Sobolev embedding, we have 1 1 d + 1 + un ≥ + − un θ p
p−
q+
− c1 un
(21)
.
Since θ > p + and 1 < p − < q + , (21) implies that {un } is bounded in E. Proposition 3.6. Assume b ( x ) satisfies (B0), f ( x ) satisfies (F0) and (F4), and let {un } be a (PS)d sequence in E, then Φ − satisfies the (PS) condition. Proof. From Proposition 3.4, we have lim
n →∞
( ∫ ( ∇u N
p( x)−2 n
p( x)−2
∇u n − ∇u
)
)
∇u ( ∇u n − = ∇u ) dx lim = ∫ N Pn dx 0.
(22)
n →∞
By Lemma 2.4, we get
lim ∫ b ( x ) un
p( x )
N n →∞
dx = ∫ N b ( x ) u
p( x )
dx.
(23)
On the other hand, Lebesgue’s dominated convergence theorem and the weak convergence of show
lim ∫ b ( x ) u
p( x )−2
N n →∞
Moreover, since b ( x )
p ( x ) −1 p( x)
un
b ( x)
p( x)−2
p( x)
un are bounded in L
p ( x ) −1 p( x)
uun dx = ∫ N b ( x ) u
un
p( x)−2
un b ( x )
p ( x ) −1
p ( x ) −1 p( x)
u
p( x )
dx.
{un }
to u in E (24)
( ) , then we have N
p( x)−2
p( x)
u in L
p ( x ) −1
( ). N
Therefore, by virtue of the definition of weak convergence, we obtain
lim ∫ N b ( x ) un
p( x )−2
n →∞
unudx = ∫ N b ( x ) u
By (23)-(25), we have
(∫ b ( x)( u lim ( ∫ ( b ( x ) u
lim
n →∞
=
n →∞
p( x)−2
N
N
n
un − u
p( x) n
p( x)−2
+ b ( x) u
)
u ( un − u ) dx
p( x)
− b ( x ) un
= 0.
1279
p( x )
dx.
(25)
)
p( x)−2
un u − b ( x ) u
p( x)−2
) )
uun dx
(26)
H. H. Qi, G. Jia
By (22) and (26), we get lim
n →∞
( ∫ ( ∇u N
p( x )−2 n
(
p( x )−2
+ ∫ N b ( x ) un
p( x )−2
un − u
)
p( x )−2
∇u n − ∇u
∇ u ( ∇ u n − ∇ u ) dx
)
)
u ( u n − u ) dx = 0.
Then combining Lemma 3.3, we obtain
lim
n →∞
(∫
N
∇u n − ∇u
p( x )
dx + ∫ N b ( x ) un − u
p( x )
)
dx = 0,
which imply that un → u in E. Proposition 3.7. There exist r > 0 and l > 0 such that Φ − ( u ) ≥ l , for all u ∈ E with u = r . Proof. The conditions (F0) and (F4) imply that p+
F ( x, t ) ≤ ε t For u
+ C (ε ) t
q( x )
, for all ( x, t ) ∈ N × .
small enough, combined with Proposition 2.3, we have Φ− (u ) ≥
(
)
1 − J1 ( u ) − ∫ N F x, u ( ) dx p+
1 u p+ 1 ≥ + u p ≥
p+
− ε ∫ N u (
−)
p+
− ε ∫ N u
p+
p+
dx − C ( ε ) ∫ N u (
dx − C ( ε ) ∫ N u
−)
q( x)
q( x)
dx
(27)
dx.
By the condition (F0), it follows
p − ≤ p ( x ) ≤ p + < q − ≤ q ( x ) < p* ( x ) , from Lemma 2.4, which implies the existence of C4 , C5 > 0 such that u
Lp
+
≤ C4 u and u q ( x ) ≤ C5 u , for all u ∈ E.
(28)
Using (28) and Proposition 2.1, we deduce
∫
( u ) dx ≤ u q( x)
N
q− q( x)
≤ C6 u
q−
.
Combining (27), it results in that
Φ− (u ) ≥
1 u p+
p+
p+
+
− ε C4p u
+
− C7 u
here Ci are positives constants. Taking ε > 0 such that ε C4p ≤
Φ− (u ) ≥
1 u 2 p+
p+
− C7 u
q−
≥ u
p+
q−
,
1 , we obtain 2 p+
1 + − C7 u 2p
q− − p+
.
− + 1 Since p + < q − , the function t → + − C7 t q − p is strictly positive in a neighborhood of zero. It follows 2P that there exist r > 0 and l > 0 such that
Φ − ( u ) ≥ l , ∀u ∈ E : u =r.
Proposition 3.8. If u ∈ E and s > 1 , we have Φ − ( su ) → −∞, as s → +∞, for a certain u ∈ E . Proof. From the condition (F3), we get
1280
H. H. Qi, G. Jia
F ( x, t ) ≥ c t , t ≥ K , for all x ∈ N . θ
For u ∈ E and s > 1 , we have
Φ −= ( su )
∫
≤ sp
N
+
(
1 s∇u p ( x)
∫
N
p( x)
(
+ b ( x ) su
p( x)
1 ∇u p ( x)
p( x)
+ b ( x) u
) dx − ∫ F ( x, ( su ) ) dx ) dx − cs ∫ u dx. (−)
N
p( x)
(−) θ
θ
N
Since θ > p + , we obtain
Φ − ( su ) → −∞, as s → +∞.
Proof of Theorem 1.2. According to the Mountain Pass Lemma, the functional Φ − has a critical point u ( − ) − 0 , that is, u ( − ) ≠ 0 , a.e. x ∈ N . Therefore, the problem ( M − ) has a nonwith Φ − u ( ) ≥ l . But, Φ − ( 0 ) = trivial solution which, by Lemma 2.8, is a non-positive solution of the problem (3). Similarly, for functional Φ + , we still can show that there exists furthermore a non-negative solution. The proof of Theorem 1.2 is now complete.
( )
Acknowledgements This work has been supported by the Natural Science Foundation of China (No. 11171220) and Shanghai Leading Academic Discipline Project (XTKX2012) and Hujiang Foundation of China (B14005).
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