- Research Article
- Open access
- Published:
Existence of Solutions for the
-Laplacian Problem with Singular Term
Boundary Value Problems volume 2010, Article number: 584843 (2010)
Abstract
We study the following -Laplacian problem with singular term:
,
,
,
, where
is a bounded domain,
. We obtain the existence of solutions in
.
1. Introduction
After Kováčik and Rákosník first discussed the spaces and
spaces in [1], a lot of research has been done concerning these kinds of variable exponent spaces, for example, see [2–5] for the properties of such spaces and [6–9] for the applications of variable exponent spaces on partial differential equations. Especially in
spaces, there are a lot of studies on
-Laplacian problems; see [8, 9]. In the recent years, the theory of problems with
-growth conditions has important applications in nonlinear elastic mechanics and electrorheological fluids (see [10–14]).
In this paper, we study the existence of the weak solutions for the following -Laplacian problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ1_HTML.gif)
where is a bounded domain,
,
is Lipschitz continuous on
, and
Let be the set of all Lebesgue measurable functions
. For all
, we denote
,
, and denote by
the fact that
.
We impose the following condition on :
(F) ,
,
and for
, there exist
such that
, whenever
.
A typical example of (1.1) is the following problem involving subcritical Sobolev-Hardy exponents of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ2_HTML.gif)
where ,
,
,
. In fact, take
,
, and
, then it is easy to verify that (F) is satisfied.
Our object is to obtain the existence of solutions in the following four cases:
(1) ;
(2) ;
(3) ;
(4) .
When , the solution of the
-Laplacian equations without singularity has been studied by many researchers. The study of problem (1.1) with variable exponents is a new topic.
The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge of variable exponent Lebesgue and Sobolev spaces. In Section 3, we prove our main results.
2. Preliminaries
In this section we first recall some facts on variable exponent Lebesgue space and variable exponent Sobolev space
, where
is an open set; see [1–4, 8, 15] for the details.
Let and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ3_HTML.gif)
The variable exponent Lebesgue space is the class of functions
such that
.
is a Banach space endowed with the norm (2.1).
For a given we define the conjugate function
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ4_HTML.gif)
Theorem 2.1.
Let . Then the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ5_HTML.gif)
holds for every and
with the constant
depending on
and
only.
Theorem 2.2.
The dual space of is
if and only if
. The space
is reflexive if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ6_HTML.gif)
Theorem 2.3.
Suppose that satisfies (2.4). Let
,
, then necessary and sufficient condition for
is that for almost all
one has
, and in this case, the imbedding is continuous.
Theorem 2.4.
Suppose that satisfies (2.4). Let
. If
, then
(1) if and only if
,
(2) if , then
,
(3) if , then
,
(4) if and only if
,
(5) if and only if
.
We assume that is a given positive integer.
Given a multi-index , we set
and
, where
is the generalized derivative operator.
The generalized Sobolev space is the class of functions
on
such that
for every multi-index
with
, endowed with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ7_HTML.gif)
By we denote the subspace of
which is the closure of
with respect to the norm (2.5).
In this paper we use the following equivalent norm on :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ8_HTML.gif)
Then we have the inequality .
Theorem 2.5.
The spaces and
are separable reflexive Banach spaces, if
satisfies (2.4).
Theorem 2.6.
Suppose that satisfies (2.4). Let
. If
, then
(1) if and only if
,
(2) if , then
,
(3) if , then
,
(4) if and only if
,
(5) if and only if
.
We denote the dual space of by
, then we have the following.
Theorem 2.7.
Let . Then for every
there exists a unique system of functions
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ9_HTML.gif)
The norm of is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ10_HTML.gif)
Theorem 2.8.
Let be a domain in
with cone property. If
is Lipschitz continuous and
,
is measurable and satisfies
a.e.
, then there is a continuous embedding
.
Theorem 2.9.
Let be a bounded domain. If
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ11_HTML.gif)
where is a constant depending on
.
Next let us consider the weighted variable exponent Lebesgue space. Let , and
for
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ12_HTML.gif)
with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ13_HTML.gif)
then is a Banach space.
Theorem 2.10.
Suppose that satisfies (2.4). Let
. If
,
, then
(1) for ,
if and only if
,
(2) if and only if
,
(3) if , then
,
(4) if , then
,
(5) if and only if
,
(6) if and only if
.
Theorem 2.11.
Assume that the boundary of possesses the cone property and
. Suppose that
, and
, for
. If
and
then there is a compact embedding
.
Theorem 2.12.
Let be a measurable subset. Let
be a Caracheodory function and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ14_HTML.gif)
where ,
,
,
is a constant, then the Nemytsky operator from
to
defined by
is a continuous and bounded operator.
3. Existence and Multiplicity of Solutions
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ15_HTML.gif)
The critical points of , that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ16_HTML.gif)
for all , are weak solutions of problem (1.1). So we need only to consider the existence of nontrivial critical points of
.
Denote by , and
the generic positive constants. Denote by
the Lebesgue measure of
.
To study the existence of solutions for problem (1.1) in the first case, we additionally impose the following conditions.
(A-1) and
,
.
(B-1) There exists a function , such that
and
for
,
.
(C-1) there exist such that
for
,
, where
and
.
(D-1) , for all
.
Theorem 3.1.
Under assumptions (F) and (A-1)–(C-1), problem (1.1) admits a nontrivial solution.
Proof.
First we show that any sequence is bounded. Let
and
, such that
and
in
. By (A-1) and (B-1),
, and
. Let
and
, then
,
,
,
and
. Let
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ17_HTML.gif)
By (B-1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ18_HTML.gif)
By (F), we get , so there exist
such that
on
. Note
, so we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ19_HTML.gif)
By Young's inequality, for , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ20_HTML.gif)
Take sufficiently small so that
.
Note that , by Young's inequality again and for
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ21_HTML.gif)
Take sufficiently small so that
.
From the above remark, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ22_HTML.gif)
As ,
and
, we have
. Since
and
, by Theorem 2.6 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ23_HTML.gif)
when and
is sufficiently large. Then it is easy to see that
is bounded in
. Next we show that
possesses a convergent subsequence (still denoted by
).
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ24_HTML.gif)
Because is bounded in
, there exists a subsequence
(still denoted by
), such that
weakly in
. By Theorem 2.11, there are compact embeddings
and
, then
in
and
. So we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ25_HTML.gif)
Hence as
.
By (F), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ26_HTML.gif)
and similarly for every ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ27_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ28_HTML.gif)
and in
and
, we obtain
and
. Similarly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ29_HTML.gif)
Because is bounded, we get
, as
. From the above remark, we conclude
, as
.
Thus , as
. Then we get
. As in the proof of Theorem
in [6, 7], we divide
into the following two parts:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ30_HTML.gif)
On , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ31_HTML.gif)
Then as
.
On , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ32_HTML.gif)
so , as
.
Thus we get . Then
in
.
From (F) and (B-1) we have , for all
,
. So we get
, for all
,
. for all
, take
, then
. Since
and
, there exists
such that
, and
, for
. Let
, such that
for
,
for
, and
in
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ33_HTML.gif)
where . So if
is sufficiently large, we obtain
.
From (F) and (C-1), we have , then
. So we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ34_HTML.gif)
Let , then
. By Theorem 2.11,
,
and
. When
is sufficiently small,
,
and
. For any
, as
, for any
, we can find
such that
and
whenever
. Take
, then
.
is an open covering of
. As
is compact, we can pick a finite subcovering
for
from the covering
. If
we define
on
. We can use all the hyperplanes, for each of which there exists at least one hypersurface of some
lying on it, to divide
into finite open hypercubes
which mutually have no common points. It is obvious that
and for each
there exists at least one
such that
. Let
, then
and we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ35_HTML.gif)
If is sufficiently small such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ36_HTML.gif)
we have .
The mountain pass theorem guarantees that has a nontrivial critical point
.
Since is a separable and reflexive Banach space, there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ37_HTML.gif)
For , denote
,
,
.
Theorem 3.2.
Under assumptions (F), (A-1)–(D-1), problem (1.1) admits a sequence of solutions such that
.
Proof.
Let . We first show that
is weakly strongly continuous. Let
weakly in
. By the compact embedding
, we have
and
a.e. on
. By the inequality
and the Vitali Theorem, we get
.
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ38_HTML.gif)
When ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ39_HTML.gif)
When ,
is bounded. So we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ40_HTML.gif)
By the compact embedding , we get
in
. So by Theorem 2.12 we obtain
, that is,
. Hence we obtain that
is weakly strongly continuous. By Proposition
in [8],
as
for
. For all
, there exists a positive integer
such that
for all
. Assume
for each
. Define
in the following way:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ41_HTML.gif)
Note that as
. Hence for
with
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ42_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ43_HTML.gif)
Note that and
. Since the dimension of
is finite, any two norms on
are equivalent, then
. If
, it is immediate that
. If
, then
. As in the proof of Theorem 3.1 we can find hypercubes
which mutually have no common points such that
and
, where
. Then we need only to consider the case:
for every
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ44_HTML.gif)
Let ,
. Let
and
. Denote
. Let
be sufficiently large such that
. There at least exists one
such that
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ45_HTML.gif)
and as
. Hence we obtain that
as
. Thus for each
, there exists
such that
for
. From Theorem 3.1
satisfies
condition. In view of (D-1), by Fountain Theorem (see [16]), we conclude the result.
In the second case, we additionally impose the following condition:
(A-2) and
.
Theorem 3.3.
Under assumptions (F), (A-2), (B-1), and (C-1) there exist such that when
, problem (1.1) admits a nontrivial solution.
Proof.
It is obvious that . Let
be such that
. Since
, there exists
and
such that
, for all
. Thus
for all
. Let
be as defined in Theorem 3.1. By (C-1),
, and
, when
. Then for any
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ46_HTML.gif)
If is sufficiently small,
.
From (F) and (C-1), we have and
. By Theorems 2.8 and 2.11, there exist positive constants
such that
,
,
. When
is sufficiently small, we have
,
, and
. As in the proof of Theorem 3.1 we can find hypercubes
which mutually have no common points such that
,
and
, where
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ47_HTML.gif)
Since , for all
, when
,
. Fix
such that
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ48_HTML.gif)
Let . When
,
. As in the proof of Theorem
in [17], denote
, we have
and
. Let
. Applying Ekeland's variational principle to the functional
, we find
such that
,
,
, and
. Thus we get a sequence
such that
and
. It is clear that
is bounded in
. As in the proof of Theorem 3.1, we get a subsequence of
, still denoted by
, such that
in
. So
and
.
Theorem 3.4.
Under assumptions (F), (A-2), and (B-1)–(D-1), problem (1.1) has a sequence of solutions such that
.
Proof.
First we show that any sequence is bounded. Let
and
, such that
and
in
. By (B-1), there exist
such that
. From (F), (A-2), and (B-1)–(D-1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ49_HTML.gif)
Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ50_HTML.gif)
By Young's inequality, for , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ51_HTML.gif)
Take ,
and
sufficiently small so that
,
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ52_HTML.gif)
Therefore by Theorems 2.6 and 2.9, we get that is bounded in
. Then as in the proof of Theorem 3.1
possesses a convergent subsequence
(still denoted by
). By Theorem 3.2, we can also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ53_HTML.gif)
As in the proof of Theorem 3.1 we can find hypercubes which mutually have no common points such that
and
, where
. Since the dimension of
is finite, any two norms on
are equivalent. Then we need only to consider the cases
and
for every
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ54_HTML.gif)
Hence as
. As in the proof of Theorem 3.2, we complete the proof.
In the third case, we additionally impose the following condition:
(A-3) , and
,
(B-3) there exist such that
for
.
Theorem 3.5.
Under assumptions (F), (A-3), and (B-3), problem (1.1) admits a nontrivial solution.
Proof.
By Young's inequality, for , we get
. By (F), we have
. Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ55_HTML.gif)
Take sufficiently small so that
. From Theorem 2.11,
. If
,
is bounded. Then we need only to consider the case
. As in the proof of Theorem 3.1 we can find hypercubes
which mutually have no common points such that
and
, where
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ56_HTML.gif)
where , and
,
. As in the proof of Theorem 3.2, we obtain that
is coercive, that is,
as
. Thus
has a critical point
such that
and further
is a weak solution of (1.1).
Next we show that is nontrivial. Let
be the same as that in Theorem 3.3. By (B-3),
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ57_HTML.gif)
If is sufficiently small,
.
In the fourth case, we additionally impose the following condition:
(A-4) and
.
Theorem 3.6.
Under assumptions (F), (A-4), and (D-1), problem (1.1) admits a sequence of solutions such that
.
Proof.
First we show that any sequence is bounded. Let
and
, such that
and
in
. Denote
and
. We have
and
.
We can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ58_HTML.gif)
By Young's inequality, for , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ59_HTML.gif)
Take and
sufficiently small so that
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ60_HTML.gif)
As in the proof of Theorem 3.5, , when
. Thus, we conclude that
is bounded in
. Then as in the proof of Theorem 3.1
possesses a convergent subsequence
(still denoted by
). By Theorem 3.2, we can also get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ61_HTML.gif)
As in the proof of Theorem 3.1 we can find hypercubes which mutually have no common points such that
and
, where
. Since the dimension of
is finite, any two norms on
are equivalent. Then we need only to consider the cases
and
for every
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ62_HTML.gif)
Hence we obtain as
. As in the proof of Theorem 3.2, we complete the proof.
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Acknowledgments
This work is supported by the Science Research Foundation in Harbin Institute of Technology (HITC200702), The Natural Science Foundation of Heilongjiang Province (A2007-04), and the Program of Excellent Team in Harbin Institute of Technology.
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Yongqiang, F., Mei, Y. Existence of Solutions for the -Laplacian Problem with Singular Term.
Bound Value Probl 2010, 584843 (2010). https://doi.org/10.1155/2010/584843
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DOI: https://doi.org/10.1155/2010/584843