On Elliptic Problem with Singular Cylindrical Potential, a Concave Term, and Critical Caffarelli-Kohn-Nirenberg Exponent ()
1. Introduction
In this paper, we consider the multiplicity results of nontrivial nonnegative solutions of the following problem ![](//html.scirp.org/file/7-7402891x5.png)
![](//html.scirp.org/file/7-7402891x6.png)
where
, where each point x in
is written as a pair
where k and N are integers such that
and k belongs to
, ![](//html.scirp.org/file/7-7402891x15.png)
![](//html.scirp.org/file/7-7402891x14.png)
![](//html.scirp.org/file/7-7402891x13.png)
is the critical Caffarelli-Kohn-Nirenberg exponent,
, ![](//html.scirp.org/file/7-7402891x18.png)
is a real parameter,
, h is a bounded positive function on
.
is the dual of
, where
and
will be defined later.
Some results are already available for
in the case
, see for example [1] [2] and the references
therein. Wang and Zhou [1] proved that there exist at least two solutions for
with
,
and
, under certain conditions on f. Bouchekif and Matallah [3] showed the
existence of two solutions of
under certain conditions on functions f and h, when
,
,
and
, with
a positive constant.
Concerning existence results in the case
, we cite [4] [5] and the references therein. Musina [5] con- sidered
with
instead of a and
, also
with
,
,
, with
and
. She established the existence of a ground state solution when
and
for
with
instead of a and
. She also showed that
with
,
,
does not admit ground state solutions. Badiale et al. [6] studied
with
,
,
and
. They proved the existence of at least a nonzero nonnegative weak solution u, satis- fying
when
and
. Bouchekif and El Mokhtar [7] proved that
ad- mits two distinct solutions when
,
with
,
, and ![]()
where
is a positive constant. Terracini [8] proved that there is no positive solutions of
with
,
when
,
and
. The regular problem corresponding to
and
has been considered on a regular bounded domain
by Tarantello [9] . She proved that, for
, the dual of
, not identically zero and satisfying a suitable condition, the problem considered admits two distinct solutions.
Before formulating our results, we give some definitions and notation.
We denote by
and
, the closure of
with respect to the norms
![]()
and
![]()
respectively, with
for
.
From the Hardy-Sobolev-Maz’ya inequality, it is easy to see that the norm
is equivalent to
. More explicitly, we have
![]()
for all
.
We list here a few integral inequalities.
The starting point for studying
, is the Hardy-Sobolev-Maz’ya inequality that is particular to the cylindrical case
and that was proved by Maz’ya in [4] . It states that there exists positive constant
such that
(1.1)
for any
.
The second one that we need is the Hardy inequality with cylindrical weights [5] . It states that
(1.2)
It is easy to see that (1.1) hold for any
in the sense
(1.3)
where
positive constant,
,
, and in [10] , if
the embedding
is compact, where
is the weighted
space with norm
![]()
Since our approach is variational, we define the functional J on
by
![]()
with
![]()
A point
is a weak solution of the equation
if it satisfies
![]()
with
![]()
![]()
![]()
Here
denotes the product in the duality
,
.
Let
![]()
From [11] ,
is achieved.
Throughout this work, we consider the following assumptions:
(F) there exist
and
such that
, for all x in
.
(H) ![]()
Here,
denotes the ball centered at a with radius r.
In our work, we research the critical points as the minimizers of the energy functional associated to the problem
on the constraint defined by the Nehari manifold, which are solutions of our system.
Let
be positive number such that
![]()
where
.
Now we can state our main results.
Theorem 1. Assume that
,
,
, (F) satisfied and
verifying
, then the system
has at least one positive solution.
Theorem 2. In addition to the assumptions of the Theorem 1, if (H) hold and
satisfying
, then
has at least two positive solutions.
Theorem 3. In addition to the assumptions of the Theorem 2, assuming
, there exists a positive real
such that, if
satisfy
, then
has at least two positive solution and two opposite solutions.
This paper is organized as follows. In Section 2, we give some preliminaries. Sections 3 and 4 are devoted to the proofs of Theorems 1 and 2. In the last Section, we prove the Theorem 3.
2. Preliminaries
Definition 1. Let
, E a Banach space and
.
i)
is a Palais-Smale sequence at level c ( in short
) in E for I if
![]()
where
tends to 0 as n goes at infinity.
ii) We say that I satisfies the
condition if any
sequence in E for I has a convergent sub- sequence.
Lemma 1. Let X Banach space, and
verifying the Palais-Smale condition. Suppose that
and that:
i) there exist
,
such that if
, then
;
ii) there exist
such that
and
;
let
where
![]()
then c is critical value of J such that
.
Nehari Manifold
It is well known that J is of class
in
and the solutions of
are the critical points of J which is not bounded below on
. Consider the following Nehari manifold
![]()
Thus,
if and only if
(2.1)
Note that
contains every nontrivial solution of the problem
. Moreover, we have the following results.
Lemma 2. J is coercive and bounded from below on
.
Proof. If
, then by (2.1) and the Hölder inequality, we deduce that
(2.2)
Thus, J is coercive and bounded from below on
.
Define
![]()
Then, for ![]()
(2.3)
Now, we split
in three parts:
![]()
![]()
![]()
We have the following results.
Lemma 3. Suppose that
is a local minimizer for J on
. Then, if
,
is a critical point of J.
Proof. If
is a local minimizer for J on
, then
is a solution of the optimization problem
![]()
Hence, there exists a Lagrange multipliers
such that
![]()
Thus,
![]()
But
, since
. Hence
. This completes the proof.
Lemma 4. There exists a positive number
such that for all
, verifying
![]()
we have
.
Proof. Let us reason by contradiction.
Suppose
such that
. Then, by (2.3) and for
, we have
(2.4)
Moreover, by the Hölder inequality and the Sobolev embedding theorem, we obtain
(2.5)
and
(2.6)
From (2.5) and (2.6), we obtain
, which contradicts an hypothesis.
Thus
. Define
![]()
For the sequel, we need the following Lemma.
Lemma 5.
i) For all
such that
, one has
.
ii) For all
such that
, one has
![]()
Proof. i) Let
. By (2.3), we have
![]()
and so
![]()
We conclude that
.
ii) Let
. By (2.3), we get
![]()
Moreover, by (H) and Sobolev embedding theorem, we have
![]()
This implies
(2.7)
By (2.2), we get
![]()
Thus, for all
such that
, we have
.
For each
with
, we write
![]()
Lemma 6. Let
real parameters such that
. For each
with
, one has the following:
i) If
, then there exists a unique
such that
and
![]()
ii) If
, then there exist unique
and
such that
,
,
,
![]()
Proof. With minor modifications, we refer to [12] .
Proposition 1 (see [12] )
i) For all
such that
, there exists a
sequence in
.
ii) For all
such that
, there exists a a
sequence in
.
3. Proof of Theorems 1
Now, taking as a starting point the work of Tarantello [13] , we establish the existence of a local minimum for J on
.
Proposition 2. For all
such that
, the functional J has a minimizer
and it satisfies:
i) ![]()
ii)
is a nontrivial solution of
.
Proof. If
, then by Proposition 1 (i) there exists a
sequence in
, thus it bounded by Lemma 2. Then, there exists
and we can extract a subsequence which will denoted by
such that
(3.1)
Thus, by (3.1),
is a weak nontrivial solution of
. Now, we show that
converges to
strongly in
. Suppose otherwise. By the lower semi-continuity of the norm, then either
and we obtain
![]()
We get a contradiction. Therefore,
converge to
strongly in
. Moreover, we have
. If not, then by Lemma 6, there are two numbers
and
, uniquely defined so that
and
. In particular, we have
. Since
![]()
there exists
such that
. By Lemma 6, we get
![]()
which contradicts the fact that
. Since
and
, then by Lemma 3, we may assume that
is a nontrivial nonnegative solution of
. By the Harnack inequality, we conclude that
and
, see for exanmple [14] .
4. Proof of Theorem 2
Next, we establish the existence of a local minimum for J on
. For this, we require the following Lemma.
Lemma 7. For all
such that
, the functional J has a minimizer
in
and it satisfies:
i) ![]()
ii)
is a nontrivial solution of
in
.
Proof. If
, then by Proposition 1 ii) there exists a
,
sequence in
, thus it bounded by Lemma 2. Then, there exists
and we can extract a subsequence which will denoted by
such that
![]()
![]()
![]()
![]()
This implies
![]()
Moreover, by (H) and (2.3) we obtain
(4.1)
where,
. By (2.5) and (4.1) there exists a positive number
![]()
such that
(4.2)
This implies that
![]()
Now, we prove that
converges to
strongly in
. Suppose otherwise. Then, either
. By Lemma 6 there is a unique
such that
. Since
![]()
we have
![]()
and this is a contradiction. Hence,
![]()
Thus,
![]()
Since
and
, then by (4.2) and Lemma 3, we may assume that
is a nontrivial nonnegative solution of
. By the maximum principle, we conclude that
.
Now, we complete the proof of Theorem 2. By Propositions 2 and Lemma 7, we obtain that
has two positive solutions
and
. Since
, this implies that
and
are distinct.
5. Proof of Theorem 3
In this section, we consider the following Nehari submanifold of ![]()
![]()
Thus,
if and only if
![]()
Firsly, we need the following Lemmas
Lemma 8. Under the hypothesis of theorem 3, there exist
,
such that
is nonempty for any
and
.
Proof. Fix
and let
![]()
Clearly
and
as
. Moreover, we have
![]()
If
for
,
for
, then there exists
![]()
where
![]()
and
![]()
and there exists
such that
. Thus,
and
is nonempty for any
.
Lemma 9. There exist M,
positive reals such that
![]()
and any
verifying
![]()
Proof. Let
, then by (2.1), (2.3) and the Holder inequality, allows us to write
![]()
where
. Thus, if
![]()
and choosing
with
defined in Lemma 8, then we obtain that
(5.1)
Lemma 10. Suppose
and
. Then, there exist r and
posi- tive constants such that
i) we have
![]()
ii) there exists
when
, with
, such that
.
Proof. We can suppose that the minima of J are realized by
and
. The geometric conditions of the mountain pass theorem are satisfied. Indeed, we have
i) By (2.3), (5.1) and the fact that
, we get
![]()
Exploiting the function
and if
, we obtain that
for
. Thus, there exist
,
such that
![]()
ii) Let
, then we have for all ![]()
![]()
Letting
for t large enough. Since
![]()
we obtain
. For t large enough we can ensure
.
Let
and c defined by
![]()
and
![]()
Proof of Theorem 3.
If
![]()
then, by the Lemmas 2 and Proposition 1 ii), J verifying the Palais-Smale condition in
. Moreover, from the Lemmas 3, 9 and 10, there exists
such that
![]()
Thus
is the third solution of our system such that
and
. Since
is odd with res- pect u, we obtain that
is also a solution of
.