Existence and Concentration of Sign-Changing Solutions of Quasilinear Choquard Equation ()
Keywords:

1. Introduction and Main Results
In this paper, we consider the following quasilinear equation of choquard type
(1)
where
is given real functions on
and
with
,
,
,
, and
is a small parameter,
is the Riesz potential defined by
and
is the Gamma function.
In the mathematical literature, very few results are known about Equation (1). Since the classical variational approach fails if
. In reference [1] , the author considered the following quasilinear equation
(2)
In fact, the natural functional associated to (2) is
which is not defined in
for a general coefficient
in the principal part. Moreover, even if
is smooth strictly positive bounded function, the corresponding energy functional is well defined in
, if
it is Gâteaux differentiable only along directions of
[1] . More recently, if
is a bounded subset of
a different approach has been developed which exploits the interaction between two different norm on
[2] [3] [4] . So, use the interaction between the norm
and the standard one on
, if
has a subcritical growth, we state that
satisfies a weaken version of the Cerami’s variant of the Palais-Smale condition in
. We note that, in general,
cannot verify the standard Palais-Smale condition, or its Cerami’s variant, as Palais-Smale sequences may converge in the
-norm but be unbounded in
[4] . For this reason, we know that there is a bounded radial solution of Equation (2) under certain assumptions.
For the Choquard equation
(3)
where
. When the potential V is a positive constant, Lieb [5] obtained the existence and uniqueness of positive radial ground states for (3), Lions [6] established the existence of infinitely many radial solutions, Ma and Zhao [7] studied the radial symmetry and uniqueness of positive ground states for (3) in higher dimension space via the method of moving planes. For more related results, we refer to [2] [8] [9] [10] [11] [12] and references therein.
In this article, we consider the existence of sign-changing solutions for the quasilinear Choquard Equation (1) by using the method of invariant sets of descending flow and the perturbation method. Byeon-Wang type penalization method [13] can be used to deal with multiple localized nodal solutions for semiclassical Schrödinger equations. Additional coercive term [14] can be used to make the perturbed functional has necessary compactness properties in changed space.
From now on, Let
be such that
(A0)
is a
Carathéodory function, i.e.
is measurable for all
,
is
for a.e.
;
(A1)
and
are essentially bounded if t is bounded, i.e.
(A2) Exists a constant
such that
(A3) Exist constants
and
such that
(A4) Exist constants
and
such that
The potential function V satisfies the following assumptions:
(V1)
and there exist constants
such that
(V2) There exists a bounded domain
with smooth boundary
such that
where
is the outer normal of
at x. according to condition (V2) that
(see [15] )
and
is a closed subset of
, without losing generality, we assume that
.
For any set
and any
, we denote
The main results of this paper are as follows:
Theorem 1.1 Assume that
satisfy conditions (A0)-(A4), the potential function V satisfies (V1) and (V2), for any positive integer k, there exists
such that if
, the problem (1) has at least k pairs of sign-changing solutions
. In addition, for any
there exist
and
such that if
, then it holds that
2. Preliminaries
We note that Equation (1) corresponding energy functional is
(4)
We use the penalization method due to Byeon and Wang [13] . Let
be a cut-off function,
for
;
for
;
and
. Define
In order to overcome the lack of compactness condition, we set the workspace as
, where
is a weighted space defined as
The corresponding norm is defined as
which
if
;
if
.
Define
Meanwhile, We add the forced disturbance term such that
has the necessary compactness property on
, introduce some auxiliary functions. Let
be a smooth, even function, such that
if
,
if
, and
is decreasing in
. For
,
,
, we define
Now, we define the perturbation functional:
(5)
where
,
,
sufficiently small, E satisfies the assumptions (V1) and (V2).
Note that for any
(6)
We will use the abstract critical point theorem to prove the existence of the infinitely variable sign solution of Equation (1). However, when verifying the conditions of the theorem, we encounter essential difficulties, that is, we can’t guarantee that the positive cone and the negative cone are flow-invariant, so we follow the idea of literature [14] [16] [17] and use truncation method to truncate the nonlocal term. First, we define the following auxiliary functions:
Define the perturbation functional:
(7)
where
.
By the definition of
, it is easy to prove that
satisfies the following properties:
Lemma 2.1 [15] For
, it holds
(1)
if
;
(2)
;
(3)
, where
.
, since
, we have
(8)
According to Hardy-Littlewood-Sobolev inequality and Sobolev inequality
Therefore, when
, there is
and
; when
and
, there is
and
. Hence, finding the solution of the Equation (4) translates into finding the critical point of
.
We describe the abstract critical point theorem in detail below [15] .
Let X be a Banach space and f be an even
functional on X. Let
are two open convex sets of X,
. Set
Assume
(I1) f satisfies the (PS) condition.
(I2)
. And assume there exists an odd continuous map
satisfying the following:
(F1) given
, there exists
such that if
,
, then
(F2)
,
.
Define
where
is the genus of symmetric sets, defined by
Assume
(Γ)
is nonempty.
We define
Theorem 2.2 ( [18] , Theorem 2.5) Assume (I1), (I2), (F1), (F2), (Γ) hold, then
(1)
,
,
(2)
, for
.
(3)
, if
.
3. Existence of Sign-Changing Critical Points
In this section, we first introduce some important properties of auxiliary functions, then prove that
satisfies the (PS) condition, and then use Theorem B to prove the existence of the critical point of
.
Lemma 3.1 ( [19] , Lemma 2.2), ( [20] , Lemma 2.1) For
(1)
;
(2) If
, then
;
If
, then
;
If
, then
; where
;
(3)
;
(4)
;
(5)
,
;
,
;
(6)
,
;
,
.
Lemma 3.2 Let
be a Palais-Smale sequence of the functional
, then
is bounded in
.
Proof: According to (7), (8) and assumptions (A4), (A2) and Lemma 3.1 (4) (3), we have
Thus, The (PS) sequence
of functional
is bounded in
.
Lemma 3.3 The embedding
↪
is compact.
Proof: Let
be the (PS) sequence of functional
,
satisfy
,
in
, by Lemma 3.2, there is a constant
independent of
such that
and
. Up to subsequences, suppose
in
and
in
. First prove
in
, for any
that
Hence
For
,
where
,
.
Lemma 3.4 For every
,
satisfies the Palais-Smale condition.
Proof: Let
be a Palais-Smale sequence of the functional
, so
and
, We will prove that
has a convergent subsequence in
. According to Lemma 3.2,
in
, then by assuming A2, A1, Hardy-Littlewood-Sobolev inequality, Hölder inequality and Lemma 3.3, there are
Thus
In the following proof process, the following basic inequalities will be used [21] .
, when
there is
(9)
when
there is
(10)
For
, by (9) we have
In addition, it follows from Lemma 3.1 (5) that
For
, by (10) we have
Similarly,
It follows from Lemma 3.1 (5) that
then
Thus
So,
in
,
satisfies the Palais-Smale condition.
Next, we will prove the existence of the critical point of the functional
by using the descending invariant set method Theorem 2.2. Before verifying the condition of Theorem 2.2, we will give a few important Lemmas:
Lemma 3.5 Let
,
, if
,
in
, and
a.e.
, then there is a subcolumn still marked as
, which satisfies
(1)
(11)
(2)
(12)
where
, as
,
.
To prove Lemma 3.5, we also need the following results:
Lemma 3.6 ( [22] , Theorem 4.2.7) Let
be a domain and
is bounded in
for some
, If
a.e.
, then
in
.
Lemma 3.7 ( [23] , Theorem 2.5) If
is bounded in
,
in
and
a.e.
, then for
(1)
(2)
(3)
Lemma 3.8 ( [24] , Theorem 2.6) Let
,
, and let
be bounded and such that, up to a subsequence, for any bounded domain
,
in
as
. Then, up to a subsequence if necessary,
a.e.
as
.
Proof of Lemma 3.5:
(1) We note that:
By the Hardy-Littlewood-Sobolev inequality, for sufficiently small
, there exists
, which makes
Using the Hardy-Littlewood-Sobolev inequality again, we have
For the above
, when
is large enough, there is
Similary, let
, let
be large enough so that
and
For
, let
. If
, then
, there is
and
. Observed,
a.e.
. By Severini-Egoroff theorem,
converges to u in
, thus
. So, for a large enough n
and
Finally, we estimate that
where
,
.
It follows from Lebesgue dominated convergence theorem that
This means that by the Hardy-Littlewood-Sobolev inequality,
Let
, then
Because
is bounded in
and
a.e.
, by Lemma 3.6,
in
, thus
in
, so
Thus, summing up, we obtain that
By the arbitrariness of
, conclusion (1) is established.
(2) According to Lemma 3.7, we have
thus
(13)
where
or
.
Similarly,
thus
(14)
where
or
.
The direct calculation of (13) + (14) is
According to Rellich theorem, there is a subcolumn, which may as well still be recorded as
, for any bounded region
, we have
in
. By Lemma 3.8,
a.e.
, from Hardy-Littlewood-Sobolev inequality it follows that
By Lemma 3.6, we have
in
, with
.
Because
, so
In a similar way, we can verify that
Therefore, conclusion (2) is established.
Let
The definition operator
,
is the only solution of the following equation.
(15)
Lemma 3.9
(1) For
,
For
,
(2) For
,
(3) For
,
For
,
Proof: (1) From
and Hölder inequality it follows that
For
, from (9) we have
similarly,
For
, from (10) we have
In addition, by Lemma 3.1 (6) and Hölder inequality, for
,
For
,
(2) Take
in (1) to get (2).
(3) From assume A1, A2, we have
For
, from (9) we have
From Lemma 3.1 (5), we have
Similarly, from (10) and Lemma 3.1 (5) can prove the case of
, so (3) holds.
Lemma 3.10 If
is bounded, then
is bounded.
Proof: By (15)
thus
is bounded.
Lemma 3.11 F is odd, well defined, and continuous on
.
Proof: From the definition of operator F, it is easy to know that F is an odd operator. Definition
Equation (15) has a unique solution
, which can be obtained by solving the minimization problem
. Since
thus G is coercive.
Let
be a minimizing sequence for the functional G,
in
. By the lower semicontinuity
so v is a solution of (15). Assume
are solutions of (15), taking
as the test function, we have
Hence
By Lemma 3.9 (3), for
for
Then
, we prove that Equation (15) has a unique solution of
.
The following proves that F is continuous, taking
in (15), we have
(16)
Suppose
in
, by Lemma 3.9 and Lemma 3.10, for
, we have
(17)
Similarly, for
, we have
(18)
By Lemma 3.10 and Hardy-Littlewood-Sobolev inequality, we obtain that
(19)
By Lemma 3.5 and Lemma 3.10, we have
(20)
Thus the right-hand side of (16) satisfies
(21)
Next, we estimate the left-hand side of (16), for
, by (9)
(22)
for
, by (10)
(23)
From (21) to (23), for
, we obtain
Therefore, F is continuous.
We verify the condition (F1) in Theorem 2.2
Lemma 3.12 If
, then
(1)
(2)
where
Proof: (1) By (15),
we have
(24)
Hence
(2) For
, by (7) (15) we have
while
By Höder inequality and Young inequality, we obtain that
Hence
(25)
In addition, by Lemma 3.9
(26)
that is
Thus
(27)
By (24) and Lemma 3.9(1), (27) and Young inequality, we have