Abstract
We are concerned with the following Schrödinger system with coupled quadratic nonlinearity
which arises from second-harmonic generation in quadratic media. Here ε > 0 is a small parameter, 2 ≤ N < 6, μ > 0 and μ > γ, P(x), Q(x) are positive function potentials. By applying reduction method, we prove that if x0 is a non-degenerate critical point of Δ(P + Q) on some closed N − 1 dimensional hypersurface, then the system above has a single peak solution (vε , wε) concentrating at x0 for ε small enough.
1 Introduction and main result
In this paper, we consider the following Schrödinger system with coupled quadratic nonlinearity
where ε > 0 is a small parameter, 2 ≤ N < 6, μ > 0 and μ > γ, P(x), Q(x) are positive potentials.
System (1.1) arises from the cubic nonlinear Schrödinger equation
which appears in the nonlinear optic theory and can be used to describe the formation and propagation of optical solutions in Kerr-type materials [6, 19]. Here ϕ is a slowly varying envelope of electric field, the real-valued parameter r and χ represent the relative strength and sign of dispersion/ diffraction and nonlinearity respectively, and z is the propagation distance coordinate. The Laplacian operator ∇2 can either be
temporal solitons with τ is the normalized retarded time, or
Also, (1.1) appears in the study of standing waves for the following nonlinear system
with the form
By contrast with the coupled Schrödinger system(1.3) with χ(2) nonlinearities, the following χ2 nonlinear Schrödinger system
has been extensively investigated. There are many interesting results about (1.4) under various assumptions of V1(x) and V2(x), one can refer to [1, 2, 3, 4, 7, 8, 12, 13, 14, 15, 17, 21, 24] and their references therein.
In recent decades, system(1.1) and its related problems have attracted a lot of attention. When ε = 1, (1.1) reduces to
Applying the finite dimensional reduction method, Wang and Zhou [22] constructed the infinitely many non-radial positive solutions of (1.5) if the potential functions P(x), Q(x) are radial and satisfy some algebraic decay at infinity. Also, if ε is small, for any positive integer k ≤ N + 1, Tang and Xie [20] proved that (1.1) has a k spikes solution concentrating at some strict local maximum of P(x) and Q(x) by using the finite dimensional reduction provided that
Here we want to mention that, very recently, Luo, Peng and Yan [13] revisited the following Schrödinger equation
with 2 < p < 2*. Under the condition that V(x) obtains its local minimum or local maximum x0 at a closed N − 1 dimensional hypersurface, they obtained the existence of a positive single peak solution for (1.6) concentrating at x0 if x0 is non-degenerate critical point of ΔV and also verified the local uniqueness of single peak solutions by using local Pohazaev type identity.
Motivated by [13, 20, 22], we want to apply the finite-dimensional reduction to study the existence of positive single peak solutions for (1.1). Our purpose here is to prove that (1.1) has a single peak solution concentrating at some non-degenerate critical point of Δ(P + Q) on a closed N − 1 dimensional hypersurface.
To state our results, throughout this paper, we assume that P(x), Q(x) obtain their local minimum or local maximum at a closed N − 1 dimensional hypersurface Γ. Without loss of generality, we suppose that P(x) = Q(x) = 1 if x ∈ Γ and more precisely, we assume that P(x), Q(x) satisfies the following conditions.
(H1)There exist δ > 0 and a closed smooth hypersurface Γ such that if y ∈ Γ, P(y), Q(y) = 1 and P(y), Q(y) > 1 (or P(y), Q(y) < 1) for any y ∈ Wδ \ Γ, where Wδ := {x ∈ ℝN , dist(x, Γ) < δ}.
(H2)The level set Γt = {x : P(x), Q(x) = t} is a closed smooth hypersurface for t ∈ [1, 1 +ϑ] (or t ∈ [1−ϑ, 1]) for some small ϑ > 0. Also, for any xt ∈ Γt and x0 ∈ Γ, there holds |νt −ν| ≤ C|xt −x0| and |ςi,t −ςi| ≤ C|xt −x0|, i = 1,2, ··· , N − 1. Here, throughout this paper, we denote by νt(ν) the outward unit normal vector of Γt(Γ) at xt(x0), while we use ςi,t(ςi) to denote the i-th principal tangential unit vector of Γt(Γ) at xt(x0).
(H3)For any x ∈ Br(x0), it holds P(x) = Q(x), where x0 ∈ Γ and r is a small positive constant.
Remark 1.1
Let
where
Let us point out that if Γ is a local minimum (or local maximum) set of P(x) and Q(x), then for any x ∈ Γ,
and
This implies that for any tangential vector ς of Γ at x, one has
where Dς denotes the directional derivative at the direction ς.
Let U be the unique positive radial solution of the following problem
It is well-known in [10] that U(x) is strictly decreasing and its s order derivative satisfies
for |s| ≤ 1 and some constant C > 0.
For xε close to x0, if we denote
then
Also, write
which is the solution of
Take
since (H3) holds.
For ε > 0 small, we will use (Vε,xε ,Wε,xε ) to construct the single peak solutions concentrating at x0. First we give the following definitions.
Definition 1.2
We say that (vε , wε) is a single peak solution of (1.1) concentrating at x0 if there exist
where
Definition 1.3
We say that a critical point x0 ∈ Γ of Δ(P + Q) on Γ is non-degenerate if it holds
The main result of this paper is the following.
Theorem 1.4
Assume that (H1)− (H3) hold. If x0 ∈ Γ is a non-degenerate critical point of Δ(P + Q), then there exists ε0 > 0 such that (1.1) has a single peak solution (vε , wε) concentrating at x0 provided ε ∈ (0, ε0].
As in [13, 20, 22], we mainly apply the finite dimensional reduction method to prove our main result. Compared with [13],we have to overcome much difficulties in the reduction process which involves some technical and careful computations due to the χ(2) nonlinearity appears. Moreover, to our best knowledge, our result exhibits a new phenomenon for the coupled Schrödinger system with χ(2) nonlinearity.
Remark 1.5
Combining the ideas from [9, 13], where in [9] the coupled nonlinear Gross-Pitaevskii system was studied, we guess that the following conclusions may hold.
(1) On the basis of Theorem 1.4, further we can prove the local uniqueness of single peak solutions by using local Pohazaev type identity.
(2) Under the conditions of Theorem 1.4, if Δ(P + Q) has an isolated maximum or minimum point x0 ∈ Γ, then for any integer k > 0, (1.1) has a k-peaks solution whose peaks cluster at x0.
The structure of this paper is organized as follows. In section 2, we do some preliminaries and then we carry out the finite dimensional reduction. We will prove our main result in section 3. In the sequel, for simplicity of notations we write
2 the finite dimensional reduction
In this section, we mainly give some preliminaries and do the finite dimensional reduction. Hereafter, for any function K(x) > 0, we define the Sobolev space
endowed with the standard norm
which is induced by the inner product
Now we define H to be the product space
and set
for i = 1, · · · , N, where
Note that the variational functional corresponding to (1.1) is
Then I ∈ C2(H, ℝ) and its critical points are solutions of (1.1).
Set
We can expand Jε(φ, ψ) as follows:
where
and
It follows from [5] that
Lemma 2.1
There exists C > 0 independent of ε such that
where
Proof. Recall that
By the direct computations, we get
and
Observe that by letting uε(x) = u(εx), it is easy to check
with
and similarly,
This completes our proof.
Lemma 2.2
There exists C > 0 independent of ε such that
Proof. For any
for some τ > 0.
Now it is easy to check that
is a bounded bi-linear functional in
Next we want to discuss the invertibility of L in
It follows from Proposition 2.2 in [22] that
Proposition 2.3
For any μ > 0 and μ > γ, (V,W) is non-degenerate for the system (2.4) in
Using the above result, we come to discuss the invertibility of L in
Lemma 2.4
There exist constants ρ > 0 and ε0 > 0 such that for all ε ∈ (0, ε0],
Proof. We argue by contradiction. Suppose that there exist εn → 0,
Without loss of generality, we can assume that
and
Then, in view of
which implies that up to a subsequence, there exists v, w ∈ H1(ℝN) such that as n → +∞,
Now we claim that v = w = 0. Considering (2.5), we find
where
and
On the other hand, being
So, taking
Also, since
which gives that by letting n → +∞,
Now we take
Meanwhile, we can decompose
Then from
But by decay property of U, there exists C > 0 such that
Hence, combining (2.10)-(2.12), we can easily check that ai,n → 0as n → +∞and then it follows from (2.6) that
This implies that for any
On the other hand, from the fact that (αU, βU) solves (2.4), we see
which, together with (2.14), yields that for any
So (v, w) is a solution of
Using Proposition 2.3, there exist bi ∈ ℝ, i = 1, 2, · · · , N such that
But (2.9) gives that bi = 0, i = 1, 2, · · · , N. That is (v, w) = (0, 0), which is exactly our claim. Finally, taking into account that
where oR(1) → 0 as R → +∞.
As a result, from (2.8) and (2.7), we deduce that
which is impossible for large n and R. So we complete this proof.
Proposition 2.5
For ε > 0 sufficiently small, there is (φε , ψε) ∈ Eε,xε such that
Moreover,
for some constant C > 0 independent of ε.
Proof. We will use the contraction mapping theorem to prove the wanted result. By Lemma 2.2,
Hence finding a critical point for Jε(φ, ψ) is equivalent to solving
It follows from Lemma 2.4 that (2.16) can be rewritten as
Now we set
with θ ∈ (0, 1). Then for any (φ, ψ) ∈ Sε,
Then A maps Sε to Sε. On the other hand, for any (φ1, ψ1), (φ2, ψ2) ∈ Sε, form Lemma 2.1,
where t ∈ (0, 1). This gives that A is a contraction map from Sε to Sε. Applying the contraction mapping theorem, we can find a unique (φε , ψε) ∈ Sε satisfying (2.17) and
Furthermore, in view of (2.17), we get
which implies that
3 Proof of our main result
In this section, we assume that x0 ∈ Γ is a non-degenerate critical point of Δ(P + Q) and we will construct a single peak solution (vε , wε) of (1.1) concentrating at x0.
From Proposition 2.5, we can get the following result.
Proposition 3.1
There exists an ε0 > 0such that for any ε ∈ (0, ε0] and y close to x0, there is (φε,y , ψε,y) ∈ Eε,y such that for any (g, h) ∈ Eε,y,
Moreover,
To get a true solution of (1.1), we need to choose y such that
where
which is exact the Pohozaev type identity.
For y close to x0, y ∈ Γt for some t close to 1. In the following, we denote by ν the unit normal vector of Γt at y and we use ςi , i = 1, · · · , N − 1, to denote the principal direction of Γt at y. Then, at y, one has
and
First, we prove the following results.
Lemma 3.2
If (H1)− (H3) hold, then
is equivalent to
Proof. By the direct computations, we have
On the other hand, using Taylor’s expansion, we get
Combining (3.3) and (3.4), we find
Lemma 3.3
Under the conditions (H1)− (H3),
is equivalent to
where B1 is some constant, K(y) is a smooth function, and G1(x) = ∇P(x), ς, G2(x) = ∇Q(x), ς.
Proof. Since for any fixed d > 0 and j = 1,2
we have
On the other hand, from Gj(y) = 0, j = 1,2, we get
As a result, the result follows.
Proof of Theorem 1.4
Now, by Lemmas 3.2 and 3.3, (3.2) is equivalent to
which is also equivalent to
Let
which, together with the non-degenerate assumption, yields that DνP(y) + DνQ(y) = O(ε2) can be written as
Let
and
where ∇T is the tangential gradient on Γ at x0 and ςi,0 is the i − th tangential unit vector of Γ at x0. Hence (DςΔP)(y) + (DςΔQ)(y) = O(ε2) can be rewritten as
So we can solve (3.7) and (3.8) to get y = xε with xε → x0 as ε → 0.
Acknowledgements
The authors sincerely thank Dr. Chunhua Wang for her helpful discussions and suggestions. This paper was partially supported by NSFC (No.11601194, 11961043, 11901249).
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Conflicts of Interest
Authors state no conflict of interest.
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Data Availability
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
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