1. Introduction and Main Results
We consider the following Kirchhoff-type equation with a varying non-local term
where
is a constant, parameters
, exponents
,
and
is a Lagrange multiplier. The
in (
1) arises as a varying non-local term.
In recent years, there have been many articles involved in different types of varying non-local problems similar to (
1) such as the model
which mainly studied the existence of solutions by using variational theory and analytical methods, as seen in [
1,
2,
3,
4].
Especially for
in (
1), the Kirchhoff-type constrained minimization problems are related to
which have attracted a significant number of mathematicians to study their existence, non-existence, uniqueness and limit behavior of constraint minimizers. More detailed, for
, Ye [
5,
6] obtained some results of existence and nonexistence on constraint minimizers. Zeng and Zhang [
7] proved the local uniqueness of minimizer, and then they [
8] provided an analysis of asymptotic behavior for minimizers when
satisfies periodic potential. Guo, Zhang and Zhou [
9] analyzed the existence and limit behavior of minimizers if the trapping potential
satisfies
. In papers [
10,
11,
12,
13], the authors studied the existence and non-existence of constraint minimizers for the Kirchhoff-type energy functional with a
-subcritical term. Also for
being a polynomial function, the articles [
14,
15] obtained the limit behavior of
-norm solutions when
as
or
as
.
Coincidentally for
and
replaced by
, the (
1) comes from an interesting physical context, which is associated with the well known Bose–Einstein condensates (BECs). The mathematical theory study of BECs can be described by a Gross–Pitaevskii (GP) functional [
16,
17], which is related to the elliptic equation
There are many researchers devoted to exploring the properties of the ground states for the GP functional related to the above elliptic equation. More precisely, when the external trapping potentials
are in the forms of polynomial, ring-shaped, multi-well, periodic and sinusoidal, the articles [
18,
19,
20,
21,
22] gave the existence, non-existence and mass concentration behavior analysis of the ground states. If
behaves like logarithmic or homogeneous potential [
23,
24], the local uniqueness and refined spike profiles of ground states for the GP functional are analyzed when
tends to a critical value
.
However, as far as we know, there are few papers using the constrained variational approaches to study the varying non-local problem (
1). Inspired by the above articles, the aim of the present paper is to study the following constrained minimization problem related to (
1), which is defined by
where
fulfills
The above
in (
2) is restricted to meet
where
satisfies
as well as with the norm
. Assume that the
in (
1) satisfies
To state our main results, we introduce an elliptic equation such as
In fact, up to the translations, (
5) has a unique positive radially symmetric solution
, as seen in [
25]. For convenience, we denote a critical constant
where
Q is the unique positive solution of (
5) for
. According to the above conditions, the existence and non-existence theorems on constraint minimizers for
are established as follows:
Theorem 1. For , and if holds, then has at least one minimizer for or . The has no minimizer for or .
Theorem 2. For , , and if holds, then has at least one minimizer if . Moreover, has no minimizer for
Remark that similar conclusions appear elsewhere for studying different types of Kirchhoff equations, as seen in [
7,
12,
14,
15]. For convenience, we give a detailed proof of Theorems 1 and 2 in
Section 3. In view of the above theorems, one knows that, for
,
and
, the
has at least one minimizer. However, for
,
and
, the
admits no minimizer. A nature question is what happens to constraint minimizers of
when
tends to 0 from the right?
Suppose that
is a minimizer for
; then, one can restrict
due to
for any
. At the same time, we always assume that
admits a positive minimizer by applying the strong maximum principle to (
1). In truth, for any positive sequence
with
as
, one can verify that the positive constraint minimizers
satisfy
as
(see
Section 4); that is, the minimizers enact blow-up behavior as
. In order to obtain a more detailed limit behavior of the constraint minimizers, some appropriate assumptions on
are necessary. For this purpose, we assume that
is a form of polynomial function, and admits
isolated minima. More narrowly, there exist
distinct points
, numbers
and constant
fulfilling
here,
exists for all
. For convenience, we denote
where
satisfies (
5) for
. Moreover, let
and the set of flattest global minima for
is denoted by
In light of Theorems 1 and 2, and inspired by [
12,
14,
15,
26], for any positive sequence
and set
being the positive minimizers of
, we next establish the following theorem on limit behavior of constraint minimizers for
when
and
as
.
Theorem 3. Assume that and hold. For , and any positive sequence with as , define ; then, the following conclusions hold:
- (i)
The has a unique local maximum satisfying and is a flattest global minimum of . Moreover, we have as where Q denotes the unique positive solution of (5) for . - (ii)
The fulfills as - (iii)
The least energy satisfies as where are stated by (6) and (7).
Notice that the
in Theorem 3 means
as
. In fact, for the case in which
and
behave in sinusoidal, ring-shaped, periodic and multi-well forms, the papers [
19,
20,
21,
22] widely studied the mass concentration behavior of the constrained minimizers.Particularly for
, the authors in [
14,
15] also analyzed the limit behavior of minimizers when
as
or
as
. As described in Theorem 3, our paper obtains an interesting result on this topic when it involves a varying non-local term, and it thus enriches the study of such issues.
The present paper is structured as follows.
Section 3 shall establish the existence and non-existence proof of constrained minimizers for
when the parameters
and exponents
satisfy suitable range. For
,
and any positive sequence
with
as
, in
Section 4 we plan to give the accurate energy estimation of
, and then analyze the detailed limit behavior of positive constrained minimizers as
.
4. Proof of Theorem 3
In this section, for , and any positive sequence with as , we plan to analyze the limit behavior on minimizers for as . Before proving Theorem 3, some indispensable lemmas are necessary, which are stated as follows:
Lemma 5. Under the assumption of Theorem 3, set and ; then, as , the and satisfy Proof. If
are positive minimizers of (
2), then
satisfies
here,
denote Lagrange multipliers. Set
where
On the contrary, we assume that
as
; then,
is bounded uniformly in
. Similar to the proof of Theorems 1 and 2 in
Section 3, one asserts that there exists a
and
has a subsequence (still denoted by
) such that as
To obtain our result, we need to prove that
as
. For this purpose, we choose a test function the same as (
14). Based on (
10), (
14) and (
15), one calculates that
and
Since
satisfies
and
, one obtains that as
It thus follows from (
9) that, for
and
as
Taking
into (
23), it yields that as
We can deduce from (
3), (
22) and (
24) that
which yields a fact that
is a minimizer of
. However, this is a contradiction since Theorem 2 shows that
has no minimizer. Thus,
holds as
.
By (
21), we just have
Since
are minimizers of
for any
, we can derive from (
11) and (
24) that as
which yields that as
It hence follows from (
21) and (
25) that as
which shows as
We have finished the proof of Lemma 5. □
Assume that
are positive minimizers of
for any
. Since
, one has
as
. This yields that
has at least one local maximum, which is denoted by
. We define a function
where
is given in Lemma 5. We next establish the following lemma, which is related to convergence properties of
and
.
Lemma 6. Under the assumption of Theorem 3, set as a local maximum of and defined by (26); then, we have - (i)
There exist a finite ball and a constant such that - (ii)
The is a unique maximum of and satisfies for some as . Furthermore, the is a minimum of , that is, .
- (iii)
The function satisfieswhere Q is the unique solution of (5) for .
Proof. (i) By (
20), we see that
fulfills the elliptic equation
here,
are Lagrange multipliers. In truth, (
2) and (
20) give that
Repeating the proof of (
24), one obtains that as
Since
yields
, we can obtain from (
28), (
29) and Lemma 5 that as
Since
take local maxima at
, it yields that
obtain local maxima at
. We thus derive from (
27) and (
30) that there exists a constant
satisfying as
Furthermore, one obtains from (
27) that
where
. In view of the De Giorgi–Nash–Moser theory, as seen in [
30] (Theorem 4.1), one declares that there exist a finite ball
and constant
such that
It hence yields from (
31) and (
33) that there exists a constant
satisfying
(ii) On the contrary, one may assume that
as
. By applying (
34) and Fatou’s lemma, for any large constant
, one has
which contradicts (
29), and it hence shows that
is bounded in
. Taking a subsequence of
if necessary (still denoted by
), there admits a
such that
as
. In fact, one can claim that
is a minimum of
, that is,
. If not, repeating the proof of (
35), it also yields a contradiction. Thus, we say that
as
and
.
(iii) The Lemma 5 shows that sequence
is bounded in
, and under the sense of subsequence, there exists a
such that
as
. Using (
30) and passing weak limit to (
27), one obtains that
satisfies
where
. By (
34) and applying the strong maximum principle to (
36), one has
. Taking
in (
5), one knows that
Due to the fact that (
37) has a unique positive radially symmetric solution
, it hence yields from (
36) that
Similar to the procedure of Theorem 1, one declares that as
,
strongly in
. Using the standard elliptic regularity theory, we obtain from (
27) that as
Applying the method [
18] (Theorem 2), one knows that the
in (
38), and 0 is the unique global maximum of
. Therefore,
behaves like
By (
39), using the technique of [
19] (Theorem 1.2), we know that
is the unique global maximum of
. □
To obtain a more detailed description on limit behavior of constraint minimizers as , some precise energy estimation of as is necessary. Toward this aim, we begin with the upper-bound estimation of , which is sated as the following lemma:
Lemma 7. Assume that and hold. If and , then for any positive sequence with as , the satisfies as where are defined by (6) and (7). Proof. Choosing (
14), we can deduce from (
9)–(
11) that there exist positive constants
such that as
and there exist positive constants
such that as
Since
satisfies
and
, we derive that there exist positive constants
such that as
where
defined by (
6) and (
7). Using (
41)–(
43), we have
Taking
, one can deduce from (
44) that as
□
Proof of Theorem 3. According to the results of Lemmas 5–7, it remains to prove
(ii) and
(iii) in Theorem 3, which can be realized by establishing the precise lower energy estimation of
as
. To meet this goal, we set
as the positive minimizers of
,
being their unique global maxima, and we define
by (
26). Using Lemma 6, one knows that for
, choosing a subsequence if necessary (still stated by
), the
and
.
In fact, we can go a step further, that is, we can come to the following conclusion:
where
and
denotes a flattest global minimum of
. To obtain (
45), we firstly claim that
If this is false, then we assume that
as
. It then follows from
and Lemma 6 (i) that, for any large positive constant
,
Recall from G-N inequality (
11) that we also have for
and
which together with (
47) then gives
where
is a arbitrarily large constant. However, this is a contradiction with the upper energy in Lemma 7. Hence, (
46) holds. In truth, the upper energy of
also compels that
. If not, by repeating the proof process from (
46) to (
48), one still derives a contradiction. Thus, we complete the proof of
(i) in Theorem 3.
Using (
45) and similar to estimation of (
47), one can deduce that there admits a
such that
where
given by (
6) and (
7). As a fact, the equality in (
50) holds only for
. One then calculates from (
49) and (
50) that
Due to the restriction of energy upper bound in Lemma 7, it yields that
is in the form of
which shows a fact that the
(ii) in Theorem 3 holds.
Taking the above
into (
51), we can obtain that
which together with Lemma 7 yields that as
So far, we have finished the proof of
(iii) in Theorem 3. □