1. Introduction
In this paper, we focus our attention on the following problem:
(1.1)
where
is a bounded domain in
,
,
and
,
,
,
and
with
here
denotes the fractional Laplace operator defined, up to a normalization factor, by
. (1.2)
The aim of this paper is to study the existence of solutions, we will see that if
, then by concentration-compactness principle, together with mini-max arguments, we can prove the existence of solutions for (1.1). We now summarize the main result of the paper.
Theorem 1.1. Let
,
and
with
. Moreover,
is bounded on
. Then
1) For any
, there exists
, then for any
, (1.1) has a consequence of weak solutions
.
2) For any
, there exist
, then for any
, (1.1) has a consequence of weak solutions
.
We denote by
the usual fractional Sobolev space endowed with the so-called Gagliardo norm
(1.3)
Then we defined
(1.4)
endowed with the norm
(1.5)
we refer to [1] for a general definition of
and its properties.
Observe that by [ [2] , Proposition 3.6] we have the following identity
(1.6)
In this work, the Sobolev constant is given by (can be seen in [ [3] , theorem 7.58])
(1.7)
where
(1.8)
2. Statements of the Result
We will use a variational approach to find a solution of (1.1). Firstly, we will associate a suitable functional to our problem, the Euler-Lagrange functional related to problem (1) is given by
defined as follow
(2.1)
To proof that J satisfy the Palais Smale condition at level c, we need the following lemma.
Lemma 2.1 [4] Letting
be a regular function that satisfies that for some
(2.2)
and
(2.3)
Let
be a bilinear form defined by
(2.4)
then, for every
, there exist positive constant
and
, such that for
, one has
and
. (2.5)
To establish the next auxiliary result we consider a radial, nonincreasing cut-off function
and
(2.6)
Lemma 2.2. [4] Letting
be a uniformly bounded in
and
the function defined in (2.6). Then,
(2.7)
Lemma 2.3. [4] With the same assumptions of Lemma 2.8 we have that
(2.8)
where B is defined in (2.4).
Lemma 2.4. [5] (Minimax principle) Assume that
, and
is a family of nonempty subset of X, denote
(2.9)
If the following conditions holds:
1) c is a finite real number;
2) there exists an
, such that
is invariant with respect to the family of mappings;
, (2.10)
that is, for any
, there holds
Then, E possesses a
sequence at level c define as (6.1.1); Furthermore, if E satisfies the
condition (or the
condition at level c), then c is a critical value of E.
3. Proof of Theorem 1.1
Firstly, recalling that J is said to satisfy the Palais Smale condition at level c if any sequence
such that
and
has a convergent subsequence.
Lemma 3.1. The
sequence
for J is bounded.
Proof. Note that
satisfies
(3.1)
and
(3.2)
where
as
. Choose
as test function in (3.2), we get that
(3.3)
therefore, by (3.1) and (3.2), we have
(3.4)
which yields the boundeness of
in
,since
.
If
, then for
, similar to the proof of
, we get
Which also yields the boundedness of
sequence
.
Lemma 3.2. Assume that
. Then
1) For any
, there exists
, such that for any
, then J satisfies
.
2) For any
there exists
such that for any
, then J satisfies
.
Proof. By Lemma3.1
is bounded in
, up to a subsequence, we get that
.
,
. (3.5)
a.e.
.
Following [6] it is easy to prove that
could also be the
-norm. Applying [ [7] , Theorem1.5], we have that the exist an index. Set
a sequence of point
and two sequences of nonnegative real numbers
, such that
. (3.6)
moreover
. (3.7)
in the sense of measures, with
for every
(3.8)
here
denotes the Dirac Delta at
, while
is the constant given in (1.7), we consider
a nonincreasing cut-off function satisfying
and
(3.9)
Set
taking the derivative of (1.6), for any
. We obtain that
(3.10)
Then, taking
as a test function in
(3.11)
by (3.10), we have
(3.12)
therefore, by (3.5) (3.6) and (3.7) we get
(3.13)
Since
is regular function with compact support, it is easy to see that it satisfies the hypothesis of Lemma 2.1, by Lemma 2.2 and Lemma 2.3 applied to the sequence
, it follows that the left hand side of (3.13) goes to zero. We obtain that
(3.14)
Clearly, if
, we get
; if
, by (3.8), we get
or
.
suppose that
, we know that
(3.15)
according to the embedded theorem, we have
(3.16)
This yields that
. (3.17)
Thus, if
, we get that
(3.18)
However, if
is given, we can choose
so small for every
that last term on the right-hand side above is greater than 0 which is contradiction when
is the same as
greater than 0. We see that
cannot occur if
or
are choose properly. Thus
. As consequence, we obtain that
in
, that is
. This implies convergence of
in
. Finally using the continuity of the inverse operator
. We obtain strong convergence of
in
. #
Next, by using the classical concept and properties of the genus, we construct a min-max class of the critical point.
For a Banach space X, We define the set
For
, define
(3.19)
If there is no mapping
as above for any
, there
. we refer to [8] for the properties of the genus.
Proposition 3.3. [8] Let
,
1) If there exists an odd map
, then
;
2) If
, then
;
3)
;
4) If S is a sphere centered at the origin in
, then
;
5) If A is compact, there exists a symmetric Neighborhood N of A, such that
.
According Holder inequality, we get that
(3.20)
We define the function
(3.21)
Then it is easy to see that given
, there exists
so small that for every
, there exists
such that
for
,
for
. and
. Analogously, for given
, we can choose
with the property that
as above for each
. Clearly,
.
As in [9] , Let
be a nonincreasing
function such that
if
and
. if
. Set
, we make the following truncation of the function J:
(3.22)
then
(3.23)
where
.
It is clear that
and is bounded from below.
Lemma 3.4. [10] 1) For any
and
or any
and
, if
, then
and
.
2) For any
, there exists such that if
and
then
satisfies
.
3) For any
,there exists
such that if
and
then
satisfies
.
Lemma 3.5. Denote
. Then for any
, there is
such that
.
Proof. Denote by
the closure of
with the respect to norm
,
in
. Extending functions in
by 0 outside
. Let
be a m-dimensional subspace of
. For any
. We write
with
and
. From the assumptions of
, it is easy to see for every
with
that there exists
such that
(3.24)
For
. Since all the norms are equivalent, we get
Therefore for given
and
. we can choose
sufficiently small so that
.#
Let
. Then
, Hence by proposition 3.3 (2) and (4)
.
We denote
and let
(3.25)
then
(3.26)
because
and
is bounded from below.
Proposition 3.6. Let
be as in Lemma 3.5 (2) and (3). Then all
given by (3.25) are critical values of
and
as
.
Proof. Denote
. Then by Lemma 3.4 (2) and (3), if
,
is compact. It is clear that
. By (3.26)
. Hence
. Moreover, since
satisfied, it follows from a standard argument (see [11] ) that all
are critical values of
. Now, we claim that
. If
because
is compact and
, it follows from Proposition 3.3 (5) that
and there exists
such that
. By the deformation Lemma [9] , there exists
and an odd homeomorphism
such that
(3.27)
Since
is increasing anad converges to
. there exists
such that
. (3.28)
And exists a
such that
(3.29)
By Proposition 3.3 (3), we obtain
(3.30)
By Proposition 3.3 (1), we obtain
(3.31)
therefore
consequently, from (3.28), we get
(3.32)
on the other hand, by (3.27) and (3.29)
(3.33)
which implies that
(3.34)
this contradicts to (3.32).Hence
. #
By (1) of Lemma 3.4
if
. This and Proposition 3.6 give Theorem1.1.