Existence and Multiplicity of Solutions for Quasilinear p(x)-Laplacian Equations in RN ()
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
-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
(1)
Also Xiaoyan Lin and X. H. Tang in [4] studied the following quasilinear elliptic equation
(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
.
In this paper, we consider the following problem
(3)
where
is the
-Laplacian operator;
is a Lipschitz continuous function with
is a given continuous function which satisfies
(B0)
here m is the Lebesgue measure on RN.
is a Carathéodory function satisfying the subcritical growth condition
(F0)
for some
, where
,
,
, and
Define the subspace
and the functional
,
where
.
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,
for all
.
If
for a.e.
, the constant function
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
(4)
This paper is to show the existence of nontrivial solutions of problem (3) under the following conditions.
(F1)
where
as given in (4).
(F2) There exist
and
, such that
(F3) There exist
and
such that
for a.e.
.
(F4)
as
and uniformly for
, with
. Here
is given in the condition (F0).
We have the following results.
Theorem 1.1. If
satisfies (B0),
satisfies (F0), (F1) and (F2), then problem (3) possesses at least one nontrivial solution.
Theorem 1.2. Assume
satisfies (B0),
satisfies (F0), (F3) and (F4), with
for a.e.
, 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
and basic properties of the variable exponent Lebesgue space
, we refer to [5] -[8] .
Let
,
. Define the variable exponent Lebesgue space:
For
, we define the following norm
Define the variable exponent Sobolev space:
which is endowed with the norm
It can be proved that the spaces
and
are separable and reflexive Banach spaces. See [9] for the details.
Proposition 2.1. [10] [11] Let
Then we have
1) For
,
;
2)
,
3)
4)
,
For
with
, let
satisfy
We have the following generalized Hölder type inequality.
Proposition 2.2. [9] [12] For any
and
, we have
We consider the case that
satisfies (B0). Define the norm
Then
is continuously embedding into
as a closed subspace. Therefore,
is also a separable and reflexive Banach space.
Similar to the Proposition 2.1, we have
Proposition 2.3. [13] The functional
defined by
has the following properties:
1)
;
2)
3)
Lemma 2.4. [13] If
satisfies (B0), then
1) we have a compact embedding
;
2) for any measurable function
with
, we have a compact embedding
. Here
means that
.
Now, we consider the eigenvalues of the p(x)-Laplacian problem
(5)
For any
, define
by
For all
, set
then
is a
submanifold of E since t is a regular value of H. Put
where
is the genus of I.
Define
We denote by
the eigenpair sequences of problem (5) such that
Define
, where
Lemma 2.5. For all
, let
be an eigenfunction associated with
of the problem (5). Then,
Proof. Let
. From the definition of
, it is easy to see that
.
On the other hand, since the functional
is coercive and weakly lower semi-continuous and
is weakly closed subset of E, there exists
such that
. Letting
, then
and
. Thus the lemma follows. ,
Lemma 2.6.
Proof. From Lemma 2.5, we have
Since
so we have
Then,
Thus we get
and
.
Similarly, if
is the eigenfunction associated with
, we get
and
. Finally, we obtain
On the other hand, it is easy to see that
Thus the lemma follows. ,
Now, we consider the truncated problem
where
We denote by
and
the positive and negative parts of u.
Lemma 2.7.
1) If
then
and
2) The mappings
are continuous on E.
Lemma 2.8. All solutions of
(resp.
) are non-positive (resp. non-negative) solutions of problem (3).
Proof. Define
where
From Lemma 2.7 and (F0),
is well defined on E, weakly lower semi-con- tinuous and C1-functionals.
Let u be a solution of
. Taking
in
we have
By virtue of Proposition 2.3, we have
, so
and
, a.e.
, then u is also a critical point of the functional Φ with critical value
.
Similarly, the nontrivial critical points of the functional
are non-negative solutions of problem
. ,
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
From (F1) we have, for any
, there is
such that
By contradiction, let
and
such that
(6)
Putting
, one has
. For a subsequence, we may assume that for some
, we have
weakly in E and
strongly in
.
Consequently,
. Let
, via the result above we have
and
Set
then,
From (6), (F1) and Lemma 2.6, we deduce that
This is a contradiction. Therefore, Φ is coercive on E. ,
Proposition 3.2. Assume
satisfies (F0) and (F2), then zero is local maximum for the functional
,
,
.
Proof. From (F2), there is a constant
such that
(7)
From (F0) and
, there exists
such that
(8)
By (7) and (8), we get
(9)
For
we have
Since
, there is a
such that
(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
on E, Φ is weakly lower semi-continuous and
, then, in order to prove
, we need to prove
. 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
and
in RN, then there are the following inequalities
Proposition 3.4. Assume (F0), and let
be a sequence such that
in E and
for all
as
, then, for some subsequences,
, a.e. in RN, as
and
for all
.
Proof. Let
and
such that
Let us denote by
the following sequence
From Lemma 3.3, we have
and
Recalling that
in E, we get
and so,
(11)
On the other hand, by (11) and
we obtain
Thus,
Combining Hölder’s inequality and Sobolev embedding, we deduce that
(12)
Let us consider the sets
From Lemma 3.3, we get
(13)
(14)
Applying again Hölder’s inequality,
(15)
where
and
Then,
(16)
From (12) and (13), we have
(17)
By (15)-(17), we obtain
(18)
(12) and (14) imply that
(19)
From (18) and (19),
a.e. in
. Because R is arbitrary, it follows that for some subsequence
.
Combined with Lebesgue’s dominated convergence theorem, we get
(20)
By (20) and
, we derive that
for all
. ,
Proposition 3.5. Assume (F0), and let
and
be a (PS)d sequence in E for
then
is bounded in E.
Proof. From (F0), we have
It is clear that
Assume that
for some
, then, by Proposition 2.3, Hölder’s inequality and Sobolev embedding, we have
(21)
Since
and
, (21) implies that
is bounded in E. ,
Proposition 3.6. Assume
satisfies (B0),
satisfies (F0) and (F4), and let
be a (PS)d sequence in E, then
satisfies the (PS) condition.
Proof. From Proposition 3.4, we have
(22)
By Lemma 2.4, we get
(23)
On the other hand, Lebesgue’s dominated convergence theorem and the weak convergence of
to u in E show
(24)
Moreover, since
are bounded in
, then we have
Therefore, by virtue of the definition of weak convergence, we obtain
(25)
By (23)-(25), we have
(26)
By (22) and (26), we get
Then combining Lemma 3.3, we obtain
which imply that
in E. ,
Proposition 3.7. There exist
and
such that
, for all
with
.
Proof. The conditions (F0) and (F4) imply that
For
small enough, combined with Proposition 2.3, we have
(27)
By the condition (F0), it follows
from Lemma 2.4, which implies the existence of
such that
(28)
Using (28) and Proposition 2.1, we deduce
Combining (27), it results in that
here
are positives constants. Taking
such that
we obtain
Since
, the function
is strictly positive in a neighborhood of zero. It follows that there exist
and
such that
,
Proposition 3.8. If
and
, we have
for a certain
.
Proof. From the condition (F3), we get
For
and
, we have
Since
, we obtain
,
Proof of Theorem 1.2. According to the Mountain Pass Lemma, the functional
has a critical point
with
. But,
, that is,
, a.e.
. Therefore, the problem
has a nontrivial solution which, by Lemma 2.8, is a non-positive solution of the problem (3).
Similarly, for functional
, 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).