Solutions to the coupled Schrödinger systems with steep potential well and critical exponent
-
Zongyan Lv
and
Zhongwei Tang
Abstract
In the present paper, we consider the coupled Schrödinger systems with critical exponent:
where d ≥ 2, β
ii
> 0 for every i, β
ij
= β
ji
when i ≠ j, λ > 0 is a parameter and
1 Introduction
Consider the following elliptic system with d ≥ 2 equations
where
where ı is the imaginary unit and
We begin by presenting the basic assumptions on the potentials V i (x) and a i :
(A 1)
(A 2) There exist M > 0 such that
(A 3) a i ∈ (−μ 1(Ω), − μ ∗(Ω)), i = 1, …, d, where μ 1(Ω) denotes the first eigenvalue of −Δ with Dirichlet boundary conditions and
This type of potentials are often referred to as steep potential well when λ is large. There are enormous investigations on non-linear Schrödinger equations and Schrödinger systems with steep potential well, for instance, [2]–[13]. For the type of muti-bump solutions, we refer to [2], [4], [8], [13]. Cao and Noussair in [4] (see also Ding and Tanaka in [8]), constructed the multi-bump solutions to the nonlinear Schrödinger equation with steep potential well under the assumptions that the bottom of the potential is composed of several connected bounded domains, if λ is large enough. Moreover, they showed that the solutions are trapped in the bottom Ω of the potentials as λ → +∞.
For the Sobolev critical case, Guo and the second author in [14] considered the muti-bump solutions for the following problem:
where N ≥ 5, V(x) ≥ 0 and its zero sets are not empty, λ > 0 is a parameter, a > 0 small such that the operator −Δ + λV(x) − a is definite. By using local mountain pass technique combining Contraction Image Principle, they constructed the multi-bump solution. Later on in [15], using a flow argument and a combination of global linking and local linking methods, Guo and the second author proved that the sign-changing bump solutions which is trapped in a neighborhood of Ω i , for λ sufficiently large, if the zero sets of V(x) have several isolated connected components Ω1, …, Ω k such that the interior of Ω i is not empty. In such case, the a > 0 is a constant such that the operator (−Δ + λV(x) − a) might be indefinite for λ large.
In recent years, the Schrödinger systems have been widely studied by many researchers, and related results can been seen in [16]–[18]. In [17], An and Yang dealt with the following weakly coupled nonlinear Schrödinger system
where
where u, v satisfying the additional condition
They proved the existence of minimizer for the system with L 2-subcritical growth (N ≤ 3). They also proved the existence results for different ranges of the coupling parameter β > 0 with L 2-supercritical growth (N = 5).
Recently, Liu, Song and Zou in [19] considered the following Schrödinger systems with Sobolev critical exponent in dimension three:
Clearly, the corresponding energy functional of (1.2) is
then the critical point of (1.3) lies on the level
where the Nehari type set
and
In this paper, based on the results of [19], we first prove that Schrödinger systems (1.1) for weakly cooperative case (β
ij
> 0 small) and for purely competitive case (β
ij
≤ 0) admits a least energy solution u
λ
which achieves
Before the statement of the main theorem, we introduce some notations first. Set
Then by the condition (A 3), for every a i ∈ (−λ 1(Ω), − λ*(Ω)), i = 1, …, d, E λ,1, …, E λ,d are the Hilbert spaces equipped with the following inner products
The corresponding norms are given by
It is easy to see that
From [20], by the assumptions (A
2), (A
3), it is easy to see that E
λ
is continuously embedded in
It is worth noting that we can choose
Throughout this paper, we always work under the following assumptions
Note that β
ij
= β
ji
, which provides a variational structure, then the solutions of (1.1) correspond to the critical points of the C
1 – energy functional
where
We call a solution is trivial if all its components are vanishing. We call a solution is semi-trivial if there exist at least one (but not all) vanishing component. We call a solution is fully nontrivial if all of its components are nontrivial. In particular, we mainly focus on the existence of least energy positive solutions, which attain the least energy positive level
Consider the Nehari type set
and the infimum of J
λ
on the set
where J
λ
is defined in (1.7). It is easy to see that
Our mian results of this paper are the following
Theorem 1.1.
Assume that (A 1), (A 2), (A 3) and (1.6) hold. There exists constant K > 0 such that
and for λ large enough, then
Remark 1.2.
For d = 1, system (1.2) reduces to the following
From [21], [22], we learn that (1.10) is the Brezis-Nirenberg problem and (1.10) has a positive least energy solution
Similarly, for d = 1, system (1.1) reduces to the following problem
where β 11 > 0. Using the method from [20], it is easy to see that there is a least energy positive solution ω λ of (1.11) which achieves
Moreover, for any sequences λ
n
→ +∞,
Theorem 1.3.
Assume that (A 1), (A 2), (A 3) and (1.6) hold and
and for λ large enough, then
In order to construct one-bump solution to system (1.1) with d = 2 equations in Section 4, we assume further that
(A
4) Ω consists of ℓ
0 components:
Remark 1.4.
By the assumption
where
Remark 1.5.
From the assumptions (A
1) − (A
3), we can see that the zero set Ω of V
i
(x)(i = 1, 2) is a bounded domain in
(A
5) For every 1 ≤ k ≤ ℓ
0, the operator −Δ + a
i
defined on
Theorem 1.6.
Assume that (A
1), (A
2), (A
4), (A
5) and (1.6) hold. For β
12 > max{β
0, β
1} (defined in (4.6)), then for any k ∈ {1, 2, …, ℓ
0} and a
1, a
2 > 0 sufficiently small, there exits Λ∗ > 0 such that for λ ≥ Λ∗, (1.1) admits a solution
This paper is organised as follows. Section 2 is devoted to the proof of Theorem 1.1. Section 3 is devoted to the proof of Theorem 1.3. Finally, we give the proof of Theorem 1.6 in Section 4.
Throughout the paper we use the notation ‖u‖ to denote the H
1-norm and |u|
p
to denote the L
p
-norm. The notation ⇀ denotes weak convergence. Capital latter C stands for positive constant, which may depend on some parameters and whose precise value can change from line to line. Let
where
For a vector
For a subset I ⊂ {1, …, d} with |I| = q, we denote the number of elements in set I by |I| and define
where
2 Least energy positive solutions for the weakly cooperative case
In this section, given I ⊆ {1, 2, …, d} with |I| = q, 1 ≤ q ≤ d, we consider the following subsystem
and define
Lemma 2.1.
Take
then for every I ⊆ {1, 2, …, d}, there holds
Proof.
By the definition of
Define
Lemma 2.2.
If
and for every
Proof.
For any
that is
which yields that
Before proceeding, we introduce some notations. For every I ⊆ {1, 2, …, d} with |I| = q, we define the matrix
Set
The following lemma shows that
Lemma 2.3.
For every I ⊆ {1, 2, …, d}, the set
The proof of this lemma follows a similar approach to that of Lemma 4.1 in [19], and for brevity, we omit it here.
Lemma 2.4.
Assume that
then we have
Moreover, the constrained critical points of J
λ,I
on
Proof.
Notice that β
ij
> 0 and
In fact, thanks to the choice of K 1, we have
Thus, by Lemma 2.2 we have
It follows that
which means that A I (u) is strictly diagonally dominant. Therefore,
and so
By Lemma 2.3 we know that the constrained critical points of J
λ,I
on
Next, we construct a Palais–Smale sequence at level
Lemma 2.5.
(Existence of Palais–Smale sequence) Assume that
Then for every I ⊆ {1, 2, …, d}, there exists a sequence
Proof.
By the definition of
where
By Lemma 2.1, we can assume that
Suppose that v
n
is the minimum eigenvalues of
where C
1 is independent on n. Note that
where
Lemma 2.6.
(See [23]) Assume that u
n
⇀ u, v
n
⇀ v in
Lemma 2.7.
Assume that there holds
then we have
where m λ,i is defined in (1.12).
Proof.
Without loss of generality, we only prove that
Step 1: We claim that there exists
where
Step 2: We claim that
This completes the proof.□
The following proposition will play a critical role in proving that
where K 1 is defined in (2.3). Then we have the following energy estimate.
Proposition 2.8.
Assume that there holds
Given I ⊆ {1, 2, …, d}, suppose that
Next, we present the proof of this proposition. Without loss of generality, we fix 1 ≤ q ≤ d − 1 and prove that
Before proving (2.9), let us firstly prove the following Lemma 2.9 and Lemma 2.10.
Lemma 2.9.
Assume that there holds
Given 1 ≤ q ≤ d − 1, if
Proof.
Notice that
Define the matrix
We claim that the matrix
Note that
By the definition of K 2 we have
Thus,
Therefore, B q (u) is strictly diagonally dominant, and so B q (u) is positive definite. It follows that there exists a constant C > 0 such that
which implies that
Lemma 2.10.
Assume that there holds
Given 1 ≤ q ≤ d − 1, if
where ω λ,i is a least energy positive solution of (1.11).
The proof of this lemma follows a similar approach to Lemma 2.8 in [19] and for brevity, we omit it here.
Proof of Proposition 2.8.
Without loss of generality, we prove that
Assume that
where
Notice that
By Lemma 2.9 we get that
We deduce from (2.12)–(2.14) that
This completes the proof of Proposition 2.8.□
Proof of Theorem 1.1
Next, we present the proof of Theorem 1.1. Recall that
then we have
Denote
where K 1 is defined in (2.3), K 2 is defined in (2.8), α is fixed in (2.15). From now on, we assume that β ij satisfies 0 < β ij < K for any i ≠ j.□
Conclusion of the Proof of Theorem 1.1
We will proceed by mathematical induction on the number of the equations in the subsystem. Set |I| = M, that is M the number of the equations in the subsystem, and M = 1, …, d. If M = 1, by Remark 1.2, we see that Theorem 1.1 is true.
We suppose by induction hypothesis that Theorem 1.1 holds true for every level
then
It is standard to see that
Denote
and by Lemma 2.6 we have
Passing to subsequence, we may assume that
Thus,
Next, we will show that all
Case 1:
On the other hand, by Lemma 2.2, we see that
which is a contradiction. Therefore, k
i
> 0, i = 1, 2, …, M + 1. Notice that
Hence,
Since
Let n → ∞, we have
Combining this with (2.21), we get
Then by the definition of K, K 4 and (2.16) we get
which implies k i > 3m λ,i . By Lemma 2.7 and (2.20) we have
that is a contradiction. Therefore, Case 1 is impossible.
Case 2: Only one component of
Without loss of generality, we assume that
that is a contradiction. Therefore, Case 2 is impossible.
Case 3: There are q components of
Without of loss generality, we may assume that
that is a contradiction. Therefore, Case 3 is impossible.
Since Case 1, Case 2 and Case 3 are impossible, then we get that all components of
which yields that
It follows from Lemma 2.1 and Lemma 2.4 that
Next, we will study the asymptotic behavior of
Claim: For every I ⊆ {1, 2, …, d}, there holds
where
and
Then there exists t = (t
1, …, t
I
), where 0 < t
i
< 1, i ∈ I such that
Thus the limit of
Assume that
Firstly, we claim that
Moreover, by assumption
This contradiction shows that
Adapting the same method, we shall proceed by mathematical induction on the number of the equations in the subsystem. Set |I| = M. If M = 1, by Remark 1.2, we see that Claim is true.
We suppose by induction hypothesis that Claim holds true for every level
By induction hypothesis we know that Proposition 2.8 is true for
And for each
Thus
Denote σ
i,n
= u
i,n
− u
i
, i = 1, 2, …, M + 1, and so σ
i,n
→ 0 weakly in
By Brézis–Lieb’s Lemma, we have
From Lemma 2.6 we have
and
Thus, we deduce that
and
Passing to subsequence, we may assume that
Thus,
Next, repeating the proceding steps, we get that all components of
which yields that
From [19], we know that
3 Least energy positive solutions for the purely competitive case
In this section, we consider the purely competitive case β
ij
≤ 0 and present the proof of Theorem 1.3. Recall the definitions of
Lemma 3.1.
Given I ⊆ {1, 2, …, d}. Assume that β
ij
≤ 0 for all i ≠ j, then there exist C
3 > 0 such that for any
Proof.
This follows directly from
□
Lemma 3.2.
Given I ⊆ {1, 2, …, d},
Proof.
The proof of this lemma can be completed by the method analogous to that used in Lemma 2.4. Recall the proof of Lemma 2.4, the key point is to show that A I (u) is strictly diagonally dominant. However, in the purely competitive case, the result is straightforward. Since β ij < 0 and u i ≢ 0, we have
which implies that A I (u) is strictly diagonally dominant in the purely competitive case. Then using the same arguments as in the proof of Lemma 2.4, we can easily carry out the proof of this lemma.□
Lemma 3.3.
Given I ⊆ {1, …, d} and for any λ > Λ0, there exists a sequence
Proof.
The proof is similar to that of Lemma 2.5, so we only sketch it. Recall the proof of Lemma 2.5, the crucial step is to show that the inequality (2.6) holds. Since β
ij
≤ 0 and
The remainder of the argument is analogous to that in Lemma 2.5.□
Lemma 3.4.
Given I ⊆ {1, …, d}, if for λ large enough and
then
Proof.
Based on Lemma 4.3, there exists a sequence
Thus,
By a standard argument,
If the assertion is false, then we may assume that some components of u are trivial. Let
As
On the other hand, for λ large enough, we see that
Combining (3.1) with (3.2) we know that
which leads to a contradiction. Therefore, u is fully nontrivial. This implies that
Hence,
Lemma 3.5.
Suppose that β ij ≤ 0 for all i ≠ j, for λ large enough, then
Proof.
We proceed to prove this statement by induction on the number of equations. For the case d = 1, this statement follows from Remark 1.2. Assume that the statement is true for every subsystem with |I| ≤ d − 1. Then the statement (3.3) reduces to
Without loss of generality, we may assume that I = {1, …, d − 1}. By Lemma 3.4 and our induction hypothesis, there exists a least energy positive solution
For simplicity, we may assume 0 ∈ Ω and
and set w ɛ (x) ≔ U ɛ (x)φ(x). where
From [24], we can see that
and
Note that by the standard regularity theory we have
To show that (3.4) holds, we need the following claim.
Claim: There exist r, R > 0 independent of ɛ and t ɛ,1, …, t ɛ,d ∈ [r, R] such that
The proof of Claim follows a similar approach to that of Lemma 4.1 in [19], and for brevity, we omit it here. Then
As
By (3.5)–(3.7) and R is independent of ɛ, we know that
and so
Since
This completes the proof.□
Conclusion of the proof of Theorem 1.3
Following directly from Lemma 3.2, Lemma 3.4 and Lemma 3.5, we get that
Next, we will show the asymptotic behavior of
Similarly, using the mathematical induction on the number of the equations in the subsystem and repeating the proceding process, we get that all components of
From [19], we know that
4 One-bump solution
In this section, we will consider the existence of one-bump solutions of the following system for λ large,
For any k ∈ {1, 2, …, ℓ}, the following problem is somehow the limit problem of (4.1)
Now, we define the variational functional by:
the corresponding Nehari manifold
and the level of
where
According to the Theorem 1.2 in [19], we know that under the condition (A3) and (1.7) and β
ij
≥ 0 for any i ≠ j, then
Define
where
It is easy to see that
Now we define the variational functional of (1.10) in Ω k by
Furthermore, according to Lemma 4.2 in [25] (or [26]), we know that there exist β
0 and β
1 such that for β
12 > max{β
0, β
1},
Then, we have the following Lemma
Lemma 4.1.
For β
12 > max{β
0, β
1}, let
Proof.
In fact, suppose
which contradicts to (4.4). We complete the proof.□
From Lemma 4.1 and (4.4), we observe that under the condition β 12 > max{β 0, β 1},
holds, where
4.1 Penalization of the nonlinearity
Let us denote
To construct the nonlinearity P, for each a > 0 we consider
where
For later purpose, let us mention that we can choose a as follows:
where
Then one can check that a critical point of Φ λ,k corresponds to a solution of the following equation
By the definition of P(u
1, u
2), we see that
4.2 Compactness of the modified functional
In this subsection, we will show that the functional Φ λ,k satisfies the Palais–Smale condition under certain energy level.
Firstly, we define the following minimax value.
where
Lemma 4.2.
Suppose that {u n } is a (P.S.) c sequence of the modified functional Φ λ,k , that is a sequence satisfying
Then there exists a positive constant Λ2 > 0 such that for any
Proof.
By direct computation, we have
and
Since {u n } is a (P.S.) c sequence, we have
where ϵ n → 0 as n → ∞. By (4.10), it is easy to see that the estimate (4.14) holds for some constant C > 0 independent of λ ≥ 0.□
Lemma 4.3.
The functional Φ λ,k (u 1, u 2) has Mountain-Pass geometry structure:
there exist two constants α and ρ 0 such that Φ λ,k (u 1, u 2) ≥ α for all
there is a e k ∈ E λ such that Φ λ,k (e 1,k , e 2,k ) < 0.
Proof.
Using (4.11) and Sobolev inequality, we have
Choosing ρ 0 small enough, we can get the conclusion (1).
On the other hand, for any given
Thus, we can take e k = (tu 0, tu 0) with t sufficiently large such that the conclusion (2) is true.□
From [23], we consider the polynomial function
It follows from the results of [19], we know that
Proposition 4.4.
Suppose that {(u
1,n
, u
2,n
)} is a
where
Proof.
By Lemma 4.2, we know that {(u 1,n , u 2,n )} is bounded. Thus there exists a subsequence of {(u 1,n , u 2,n )} (still denoted by {(u 1,n , u 2,n )}) such that
where i = 1, 2. Then by standard arguments, we can see that
Next we show that u
i,n
→ u
i
strongly in E
λ
, i = 1, 2. Let v
i,n
= u
i,n
− u, i = 1, 2, it follows from the Brezis–Lieb’s Lemma that
Hence it is sufficient to prove that v
i,n
→ 0 strongly in E
λ
, i = 1, 2. We show this by contradiction. Without loss of generality, up to a subsequence, we assume on the contrary that
which indicates that
By Lemma 4.2, we can see that
According to the condition (A2), we can choose
where C is a positive constant independent of λ and large n. Thus, for λ large, using (4.14), we have
where o λ (1) denotes a quantity that tends to 0 as λ → ∞. For λ large, it follows that
which implies that
By (4.15) and we set
which implies that
By (1.5) and (4.17), then for λ large, we have
It follows that
According to Lemma 4.3 and Proposition 4.4, we have proved the following existence result which is the main gredient of this subsection.
Proposition 4.5.
For any k ∈ {1, 2, …, ℓ}, there exists a critical point
4.3 Asymptotic behavior of the one-bump solutions
In this subsection, we study the asymptotic behavior of one-bump solutions
Proposition 4.6.
Suppose
Proof.
For any sequence
where ‖ ⋅‖ denotes the norm of
It follows that
which indicates that
here we use the fact that
By the definition of H(x, u 1, u 2), we know that for k ∈ {1, 2, …, ℓ} satisfies (4.2). For j ≠ k, setting (ϕ 1, ϕ 2) = (u 1, u 2) we have
We have
where
There is a contradiction. Thus (u 1, u 2) ≡ (0, 0) in Ω j for j ≠ k.
In the following, we will show that
which implies that
By the definition of
which indicates that
At last, we come to show that,
Since
Note that
which implies that
Combining with (4.19), we obtain that
Similar to (4.18), for λ large, we have
Thus we get
4.4 L ∞ estimate of the critical points for modified functional
In this subsection, we will show that the critical point
Proposition 4.7.
Fix M* > 0, there is a constant δ > 0 such that for any critical points
which implies that there is a Λ* > 0 such that for
The similar arguments can be found in the paper by (Guo and Tang [14]), for the completeness, we give the details of the proof. Before giving the proof of Proposition 4.7, we firstly present an L
∞ estimate for the solutions
Lemma 4.8.
Suppose
Proof.
Firstly, by Proposition 4.6, it is easy to see
Hence, for a small number η 0 > 0 (which we will be specified later), we have, for λ large enough,
Now we are ready to use Moser’s iteration argument to obtain the desired estimates. Let ψ denote a smooth cut-off function and γ > 1 an arbitrary number, both of them will be specified later. Multiply (4.11) by
By a direct computation, we have
On the other hand, by Hölder’s inequality and Young’s inequality, we have
Note that for any
since V i (x) ≥ 0 (i = 1, 2), it deduces that
By Sobolev imbedding theorem, we have
Combining with (4.23) and (4.24), we get
Now for
with
and
Similar to (4.26) and (4.27), we have
and
Take γ = 5 and η
0 > 0 is such that
which implies that for any
In the following, we will use above estimates combining with Moser’s iteration argument to prove (4.20). Let
where ψ is a cut-off function supported in B
2r
(y) with
Using Hölder’s inequality again, we notice that
and
where
Thus, for any ɛ > 0, we have
where i = 1, 2 . It follows from (4.31), we have
where
where
i.e.
where h = 1 + γ, R = max{r, 1} and
Then we can rewrite (4.34) in terms of N(⋅, ⋅):
where
Let m → ∞, we have
where
Since the above estimate is independent of
A direct result of the arguments in the proof of Lemma 4.8 is the following exterior Harnack-type inequality:
Lemma 4.9.
Suppose all the assumptions of Lemma 4.8 are satisfied. Then for any
Now we are ready to present the proof of Proposition 4.7.
Proof of Proposition 4.7.
By the proof of Proposition 4.5, one can find a constant C* > 0 independent of λ such that for λ ≥Λ*, we have
On the other hand, by the assumption on V
i
(x), there is a positive number
which implies that
And we may assume that
Note that for any
where α ∈ (0, 1) is such that 6 = 2α + (1 − α)q i.e.,
where ω
3 is the volume of the unit ball B
1(0). Combining (4.37)–(4.39), we obtain that, for any
where
which implies that
□
Proof of Theorem 1.6
According to Proposition 4.4, Proposition 4.5, Proposition 4.6 and Proposition 4.7, we complete the proof.□
Acknowledgments
The author Zongyan Lv thanks Chengxiang Zhang and Qihan He for the valuable discussions when preparing the paper. The authors also wound like to thank the referees for many valuable comments helping to improve the paper.
-
Research ethics: Not applicable.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Competing interests: The author states no conflict of interest.
-
Research funding: None declared.
-
Data availability: Not applicable.
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Articles in the same Issue
- Frontmatter
- Research Articles
- Solitons to the Willmore flow
- Solutions to the coupled Schrödinger systems with steep potential well and critical exponent
- Normalized solutions for nonlinear Schrödinger systems with critical exponents
- Multiplicity of normalized solutions for nonlinear Choquard equations
- Well-posedness of damped Kirchhoff-type wave equation with fractional Laplacian
- On Bobkov-Tanaka type spectrum for the double-phase operator
- Hopf’s lemma for parabolic equations involving a generalized tempered fractional p-Laplacian
- Disc stackings and their Morse index
- Singular Trudinger–Moser inequalities for the Aharonov–Bohm magnetic field
- A topological analysis of p(x)-harmonic functionals in one-dimensional nonlocal elliptic equations
- Generalized Gaussian curvature flows related to the Orlicz Gaussian Minkowski problem
-
Qualitative properties of two-end solutions to the Allen–Cahn equation in
- Radial symmetry, monotonicity and Liouville theorem for Marchaud fractional parabolic equations with the nonlocal Bellman operator
- Local analytic integrability for a class of degenerate planar vector fields
- Nonexistence and existence of solutions for a supercritical p-Laplacian elliptic problem
Articles in the same Issue
- Frontmatter
- Research Articles
- Solitons to the Willmore flow
- Solutions to the coupled Schrödinger systems with steep potential well and critical exponent
- Normalized solutions for nonlinear Schrödinger systems with critical exponents
- Multiplicity of normalized solutions for nonlinear Choquard equations
- Well-posedness of damped Kirchhoff-type wave equation with fractional Laplacian
- On Bobkov-Tanaka type spectrum for the double-phase operator
- Hopf’s lemma for parabolic equations involving a generalized tempered fractional p-Laplacian
- Disc stackings and their Morse index
- Singular Trudinger–Moser inequalities for the Aharonov–Bohm magnetic field
- A topological analysis of p(x)-harmonic functionals in one-dimensional nonlocal elliptic equations
- Generalized Gaussian curvature flows related to the Orlicz Gaussian Minkowski problem
-
Qualitative properties of two-end solutions to the Allen–Cahn equation in
- Radial symmetry, monotonicity and Liouville theorem for Marchaud fractional parabolic equations with the nonlocal Bellman operator
- Local analytic integrability for a class of degenerate planar vector fields
- Nonexistence and existence of solutions for a supercritical p-Laplacian elliptic problem
