1. Introduction and Main Results
Over the past few decades, considerable attention has been given to studying the following elliptic problem:
where
is a smooth and bounded open set of
with
,
is a positive real number, and
.
Problem
is a widely recognized example found in numerous applied scientific fields. For example, it can be interpreted as the stationary problem within a chemotaxis model [
1,
2], or as a shadow system derived from a reaction–diffusion framework in morphogenesis [
3]. Additionally, from a mathematical perspective, problem
is of particular interest because its solutions frequently exhibit the bubbling phenomenon. This refers to the emergence of concentration peaks around one or more points within the domain or on its boundary, with the solutions remaining negligibly small in other regions.
A substantial body of research has investigated the case where the exponent
q is fixed, and
is treated as a parameter. In some studies,
is assumed to converge to zero, while in others, it is considered in the limit as it goes to infinity. When
q is fixed and subcritical (i.e.,
), the only solution to problem
for small
is the constant solution. However, as
increases, non-constant solutions emerge, which exhibit blow-up at one or more points as
[
4]. For large
, the least-energy solution blows up at a boundary point where the mean curvature is maximized [
4,
5,
6,
7]. Several studies, including [
4,
8,
9,
10,
11], have examined higher-energy solutions of
that exhibit this asymptotic profile, with blow-up occurring at either boundary or interior points as
. When
q is critical (i.e.,
), the situation is significantly different. For
and small
, problem
admits non-constant solutions [
12,
13,
14]. However, the limiting equation of problem
, arising when studying the asymptotic behavior of the least-energy solution as
, has no solutions. Nevertheless, least-energy solutions
still exist for large
, and concentration phenomena appear in the following form [
15,
16]:
where, for
and
,
represents the standard bubble defined by
and these are the only solutions [
17] to the following problem:
Several studies, including [
15,
16,
18,
19,
20,
21,
22,
23,
24,
25] and the references therein have constructed higher-energy solutions of
with concentration at the boundary as
. In contrast to the subcritical case, these solutions require at least one blow-up point to be located on the boundary [
26].
Another interesting direction of research for problem
involves investigating blow-up phenomena by fixing
while allowing the exponent
q to approach the critical exponent, i.e.,
, where
is a small positive parameter. This was initially explored by Rey and Wei [
27,
28]. For
and
, they demonstrated the existence of a solution that blows up at a boundary point where the mean curvature is maximized [
27]. They also established the existence of a solution that blows up at a boundary point where the mean curvature is minimized when
and
is not convex [
27]. In dimension 3, they identified a solution with a single interior blow-up point [
28]. More recently, it was shown that for
and
, no solutions exist that exhibit blow-up solely at interior points when
is a small positive number [
29]. Additionally, in [
30], the authors extended the problem by replacing the constant
with a function
V and studied the scenario when
, constructing interior bubbling solutions. In this case, the interior blow-up points of these solutions converge, as
, to the critical points of the function
V. More recently, in [
31], the authors examined the case where a function
g is introduced in front of the nonlinear term, considering the following problem
where
is a smooth bounded domain in
,
,
is the critical Sobolev exponent for the embedding
,
is a small positive parameter, and
g and
V are
positive functions defined on
.
Assuming that the critical points of
g are non-degenerate, the authors demonstrated that, in contrast to the case where
studied in [
30], problem
does not admit interior bubbling solutions with clustered bubbles. However, they were able to construct solutions
to
with isolated interior multiple blow-up points. By “clustered bubbles”, we refer to a sum of bubbles that have rates of the same order, with their concentration points converging to a single point
y. In this scenario, it holds that
On the other hand, by “isolated bubbles”, we refer to a sum of bubbles with concentration points satisfying for each .
The results mentioned above raise a natural question: what happens when the critical points of
g are degenerate, particularly when
g satisfies a specific ’flatness’ condition? The aim of this paper is to address this question. Throughout the paper, we assume that
g satisfies a certain flatness condition (see Equation (
2)) for the precise statement). First, we provide a complete description of the asymptotic behavior of interior single-peaked solutions. Second, we show that, unlike the case where the non-degeneracy assumption is made, as studied by the authors in [
31], interior multipeaked solutions with clustered bubbles do indeed exist. We will also prove that solutions with isolated bubbles continue to exist. These existence results hold in both cases, whether the concentration points are critical points of the function
V or not.
To present our results, we need to establish some definitions. We say that a function
f satisfies a flatness condition near a critical point
y if it can be expressed as
where
h satisfies
Additionally, we say that f satisfies assumption if the following holds:
Hypothesis 1. For , the equation Note that if
y is a non-degenerate critical point of
f, then Equation (
2) holds immediately with
and the assumptions given in Equation (
3) are satisfied. Furthermore, as an example of functions
h that satisfy these assumptions, we can take
for any
.
For
and
, we define the following projection
:
The functions serve as the appropriate approximate solutions, and in their neighborhood, we find the true solution to the problem.
Now, we begin by analyzing the asymptotic behavior of solutions to that blow up at a single interior point as . We provide a complete description of the single interior blow-up scenario for solutions that weakly converge to zero. Specifically, we prove the following.
Theorem 1. Let and let be a family of solutions of having the formThen, the concentration point converges to a critical point y of g. Furthermore, if Equation (2) is satisfied in the sense of , then the concentration rate satisfieswhere , for ,Additionally, if there exists a positive constant such that in a neighborhood of y, then the concentration point satisfies the following: is bounded and two cases may arise: If there exists a positive constant β such that , then .
If , then y has to be a critical point of V.
Before presenting the rest of our results, we note that the dimension is excluded from our work, due to the following reasons:
For
, the function
and satisfies
For
, the previous integral diverges. But, since
is bounded, for
, we have
For
, this integral depends on the geometry of
, and precise computations are required to determine the principal part. In fact, we have
where
depends on
a and
. Hence, this case must be considered separately.
Next, our objective is to establish the converse of Theorem 1. More precisely, we aim to construct interior multipeaked solutions for problem
. According to Theorem 1, we see that such a construction must be centered around a critical point
y of
g. Furthermore, when the flatness condition Equation (
2) on the function
g holds in the
sense, there are two distinct cases to consider: one where the gradient of
V at
y is non-zero, and the other where it is zero. Roughly speaking, in the first case, the rate of convergence is on the order of
, while in the second case, it is negligible in comparison to
. We begin by constructing interior multipeaked solutions in the first case, and specifically, we prove the following.
Theorem 2. Let and let be l critical points of g such that, near each , the function g satisfies Equation (2) in the sense of with . Assume that for and that condition Equation (3) is satisfied. Then, for any integer , there exists a small positive real number such that for every , the problem admits a solution which develops exactly one bubble at each point for and weakly converges to zero in . More precisely, there exist values , …, satisfying Equation (7) and points as for all j such thatAdditionally, for all j, converges to a solution of the equationwhere and are the constants defined in Theorem 1, , and Theorem 2 provides the following multiplicity result concerning the number of critical points of g that are not critical points of V.
Theorem 3. Under the assumptions of Theorem 2, there exists an such that for , problem admits at least solutions, where l denotes the number of critical points of g that are not critical points of V.
Next, we consider the case where the functions g and V share common critical points. Specifically, we have the following.
Theorem 4. Let and let be m common critical points of g and V. We assume that these critical points are non-degenerate for V, and that near each such point, g takes the form.Then, for any integer , there exists a small such that for every , the problem admits a solution satisfying the following: develops exactly one bubble at each point for and weakly converges to zero in as . More precisely there exist values ,…, of order and points as for all j such thatIn addition, we have as for all . Theorem 4 provides the following multiplicity result related to the number of common critical points of g and V.
Theorem 5. Under the assumptions of Theorem 4, there exists an such that for , problem admits at least solutions, where m denotes the number of common critical points of g and V.
The constructions of interior multipeaked solutions we presented in Theorems 2 and 4 can be combined to yield solutions that concentrate at interior points, which divide into two blocks: one consisting of common critical points of g and V, and the other consisting of critical points of g that are not critical points of V. This leads to the following result.
Theorem 6. Let and let be critical points of g such that, near each points for , g takes the form Equation (2) in the sense of with . Assume that for and that Equation (3) holds. Assume further that are non-degenerate critical points V, and that near each point for , g takes the form Equation (8). Then, for any and , there exists a small such that for every , the problem admits a solution satisfying the following:with the values ’s and ’s, which are of order and the points as for all j. In addition, for all j, , and converges to a solution of the equationwhere , and are the constants defined in Theorem 2. Note that, up to this point, all of our existence theorems have dealt with interior blowing-up solutions featuring isolated bubbles. The goal of the next results is to construct interior bubbling solutions
with clustered bubbles. As mentioned, this is the key difference between the case studied in [
31], where the critical points of
g are non-degenerate, and our case, where the critical points are degenerate. In [
31], the authors demonstrated that there are no interior blowing-up solutions with clustered bubbles. In contrast, we will prove here that such solutions do exist.
To achieve this, we introduce the following notation. For
and
y, a common critical point of both
g and
V, we define the function
where
such that
if
.
Our result is stated as follows.
Theorem 7. Let and let y be a common critical point of g and V. We assume that y is non-degenerate for V and that g satisfies Equation (8) with . Let with and assume that the function has a non-degenerate critical point . Then, for any integer , there exists a small such that for every , the problem admits a solution satisfying the followingwith the values , …, which are of the order , and the concentration points satisfywhere is a small positive real, and σ is the constant defined in Equation (142). Moreover, if for each N, has a non-degenerate critical point, then the problem admits an arbitrary number of non-constant distinct solutions provided that ε is sufficiently small.
Theorem 8. Let , , and let satisfy the assumptions stated in Theorem 7. For , if , we assume that the function has a non-degenerate critical point . Then, there exists a small such that for every , the problem admits a solution satisfying the followingwith the values which are of the order , and for each , the concentration points satisfy Equation (10) when , and the following property when : To prove our results, we perform a refined asymptotic expansion of the gradient of the associated functional and subsequently test the equation using vector fields. This allows us to derive relationships between the concentration parameters. By carefully analyzing these relationships, we obtain our results. Furthermore, no assumption regarding the symmetry of the domain is required. Note that traditional blow-up analysis methods typically depend on detailed pointwise -estimates and the frequent utilization of Pohozaev identities. In contrast, the approach outlined in this paper deviates from these conventional techniques. We argue that our method, which bypasses the need for pointwise estimates and Pohozaev identities, holds significant promise for addressing non-compact variational problems that involve more intricate blow-up behaviors. Indeed, the occurrence of non-simple blow-up points adds complexity to the derivation of pointwise -estimates, rendering this process particularly challenging.
The remainder of this paper is organized as follows: In
Section 2, we provide precise estimates for the infinite-dimensional part of
.
Section 3 is dedicated to the expansion of the gradient of the functional associated with problem
. In
Section 4, we study the asymptotic behavior of solutions to
that blow up at a single interior point as
, leading to the proof of Theorem 1.
Section 5 focuses on the construction of interior blowing-up solutions with isolated bubbles, which are used to prove Theorems 2, 3, 4, 5, and 6. In
Section 6, we construct interior blowing-up solutions with clustered bubbles, thereby proving Theorems 7 and 8.
Section 7 explores possible avenues for future research. Finally, in
Appendix A, we collect several estimates used throughout the paper.
6. Construction of Interior Blowing up Solutions with
Clustered Bubbles
This section is devoted to the proof of Theorems 7 and 8. We begin by proving Theorem 7. Let
, and
y be a common critical point of
g and
V. We assume that
y is non-degenerate for
V and that
g satisfies Equation (
8) with
. Let
with
and assume that the function
has a non-degenerate critical point
where
is defined by Equation (
9). The proof strategy for Theorem 7 follows the same approach as that of Theorem 4. We begin by introducing a neighborhood of the desired constructed solutions. Let
where
is defined by Equation (
17),
Here, , , and denote the constants defined in Proposition 8 and 9.
Following the approach in the proof of Theorem 4, we reduce the problem to a finite-dimensional system. Using Proposition 2, we can achieve this reduction by identifying
that satisfies Equation (
129). Consequently, we seek a solution
that satisfies the system defined by Equations (
130)–(
132), where
is defined by
Following a similar approach as in the proof of Theorem 4, we introduce the change of variables as follows:
where
,
, and
are defined in Proposition 8. With these variable changes, it becomes straightforward to observe that
where
Subsequently, by applying Propositions 2, 7–9, we can conclude that the following estimates are valid.
Lemma 11. For ε small, the following statements hold:where , , and are defined in Propositions 7, 8 and 9, respectively. At this point, following the argument used in the proof of Lemma 9, we deduce that the constants , , and that appear in equations , , and satisfy the following estimates:
Lemma 12. Let . Then, for ε small, the following statements hold: Next, we seek to express the equations , , in a more simplified form.
Lemma 13. For ε small, equations , , are equivalent to the following system:where Proof. First, using Proposition 7, Lemma 11, Equation (
140) and the fact that
, we see that
is equivalent to the first equation of the system
.
Second, using Lemma 12, we write
Using Proposition 8 and Lemma 11, we obtain
This implies that is equivalent to the second equation of the system .
To deal with the third equation
, we write
But, combining Equation (
146), Proposition 9 and Lemma 11, Equation (
148) becomes
where
Combining Equation (
149) and Equation (
150), we obtain
Combining Equation (
151)–(
153) and the fact that
…
is a critical point of
, we see that equation
is equivalent to
which completes the proof of Lemma 13. □
At this point, we are prepared to prove the results concerning the construction of clustered bubbling solutions.
Proof of Theorem 7. Note that the system
, …,
is equivalent to
where
defined by
As in the proof of Theorem 4, we define a linear map by taking the left-hand side of the system defined by
,
and
. Since
is a non-degenerate critical point of
, we deduce that such a linear map is invertible and arguing as in the proof of Theorem 4, we derive that the system
has a solution
for
small. This implies that
admits a solution
and by construction, Equation (
10) is satisfied. The proof of Theorem 7 is thereby completed. □
Proof of Theorem 8. Let
,
,
and let
satisfy the assumptions stated in Theorem 7. For
, we assume that
and the function
has a non-degenerate critical point
. We introduce the following set
We need to construct solution in the form
We remark that, for
,
such that
and
, we have
and thus
Therefore, Propositions 7–9 can be rewritten as
Proposition 11. Let and . It holds that
This means that only the indices in the same bloc are involved in the expansions. That is the contribution of the other blocs are taken in the remainder term.
As in the proof of Theorem 7, we derive the following.
Lemma 14. For ε small, is a critical point of if and the only if the following system is satisfied where, for and ,where is defined by Equation (142) with instead of y and μ is defined in Equation (142). The existence of a solution for the system follows as in the previous proofs by using the fact that for each , the point is a non-degenerate critical point of . Following the end of the proof of Theorem 2, the constructed solution is positive. Hence the proof of Theorem 8 is completed. □