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BY 4.0 license Open Access Published by De Gruyter August 25, 2021

Existence of single peak solutions for a nonlinear Schrödinger system with coupled quadratic nonlinearity

  • Jing Yang EMAIL logo and Ting Zhou

Abstract

We are concerned with the following Schrödinger system with coupled quadratic nonlinearity

ε2Δv+P(x)v=μvw,xRN,ε2Δw+Q(x)w=μ2v2+γw2,xRN,v>0,w>0,v,wH1RN,

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.

MSC 2010: 35J10; 35B99; 35J60

1 Introduction and main result

In this paper, we consider the following Schrödinger system with coupled quadratic nonlinearity

(1.1) ε2Δv+P(x)v=μvw,xRN,ε2Δw+Q(x)w=μ2v2+γw2,xRN,v>0,w>0,v,wH1RN,

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

(1.2) iϕz+r2ϕ+χ|ϕ|2ϕ=0,

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 2τ2 for

temporal solitons with τ is the normalized retarded time, or 2=i=1N2xi2 , where x = (x1, · · · , xN) is in the direction orthogonal to z. Solitary wave solutions to (1.2) and its generations have been studied in [4, 18].

Also, (1.1) appears in the study of standing waves for the following nonlinear system

(1.3) iϕ1t=ε2Δϕ1+(P(x)+μ)ϕ1μϕ1ϕ2,(x,t)RN×R+,iϕ2t=ε2Δϕ2+(Q(x)+μ)ϕ2μ2ϕ12γϕ22,(x,t)RN×R+,

with the form ϕ1(x,t)=v(x)eiμt,ϕ2(x,t)=w(x)eiμt, where i is the imaginary unit and ε is the Planck constant. When ε = 1 and γ = 0, the existence of ground state solutions of (1.3) was proved in [27]. Besides, (1.1) is closely related to the general parabolic system with coupled nonlinearity and the nonlinear evolution equations. For this information, we can refer to [11, 16, 23, 25, 26] and references therein.

By contrast with the coupled Schrödinger system(1.3) with χ(2) nonlinearities, the following χ2 nonlinear Schrödinger system

(1.4) iϕ1t=ε2Δϕ1+V1(x)ϕ1μ1ϕ12ϕ1βϕ22ϕ1,(x,t)RN×R+,iϕ2t=ε2Δϕ2+V2(x)ϕ2μ2ϕ22ϕ2βϕ12ϕ2,(x,t)RN×R+

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

(1.5) Δv+P(x)v=μvw,xRN,Δw+Q(x)w=μ2v2+γw2,xRN,v>0,w>0,v,wH1RN.

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 kN + 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 |P(x)P(y)|L1|xy|θ1 and |Q(x) − Q(y)| ≤ L2|xy|θ2 for some positive constants L1, L2, θ1, θ2.

Here we want to mention that, very recently, Luo, Peng and Yan [13] revisited the following Schrödinger equation

(1.6) ε2Δu+V(x)u=up1,uH1RN

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 xΓ 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 yΓ, P(y), Q(y) = 1 and P(y), Q(y) > 1 (or P(y), Q(y) < 1) for any yWδ \ 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 xtΓt and x0Γ, there holds |νtν| ≤ C|xtx0| and |ςi,tςi| ≤ C|xtx0|, 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 xBr(x0), it holds P(x) = Q(x), where x0Γ and r is a small positive constant.

Remark 1.1

Let F(x)=i=1Nxi2ai21 with ai>0,aiaj(ij) and Γ=xRN:F(x)=0. Take

P(x)=Q(x)=F2+1, in W0,

where W0=xRN:i=1Nxi2ai21δ0 for some small fixed δ0 > 0. Then we can use the above conditions to the potentials P(x), Q(x).

Let us point out that if Γ is a local minimum (or local maximum) set of P(x) and Q(x), then for any xΓ,

P(x)=1,P(x)=0

and

Q(x)=1,Q(x)=0.

This implies that for any tangential vector ς of Γ at x, one has

DζP(x)=0,DςQ(x)=0,xΓ,

where Dς denotes the directional derivative at the direction ς.

Let U be the unique positive radial solution of the following problem

(1.7) Δu+u=u2,u>0, in RN,u(0)=maxxRNu(x),u(x)H1RN.

It is well-known in [10] that U(x) is strictly decreasing and its s order derivative satisfies

DsU(x)e|x||x|1N2C

for |s| ≤ 1 and some constant C > 0.

For xε close to x0, if we denote

Vˉε,xε=PxεUPxεxxεε,

then Vˉε,xE solves

(1.8) ε2Δv+Pxεv=v2,v>0,xRN,vxε=maxxRNv(x),vH1RN.

Also, write

Wˉε,xε=QxεUQxExxEε,

which is the solution of

(1.9) ε2Δw+Qxεw=w2,w>0,xRN,wxE=maxxRNw(x),wH1RN.

Take Vε,xε,Wε,χε=αVˉε,χε,βWˉε,xε with α=1μ2(μγ)μ,β=1μ. Then Vε,Xε,Wε,xε solves

(1.10) ε2Δv+Pxεv=μvw,xRN,ε2Δw+Qxεw=μ2v2+γw2,xRN,v>0,w>0,v,wH1RN

since (H3) holds.

For ε > 0 small, we will use (Vε, ,Wε, ) 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 xεRN with |xεx0| = o(1) such that

vεVε,Xε,wεWε,Xεε=oεN2,

where (u,v)ε=RNε2|u|2+P(x)u2+ε2|v|2+Q(x)v2dx

Definition 1.3

We say that a critical point x0Γ of Δ(P + Q) on Γ is non-degenerate if it holds

2Px0+Qx0v20,det2ΔPx0ςiςj1i,jN1+2ΔQx0ςiςj1i,jN10.

The main result of this paper is the following.

Theorem 1.4

Assume that (H1)− (H3) hold. If x0Γ 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 x0Γ, 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 f to mean the Lebesgue integral of f (x) in ℝN.

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

Hε,K1=uH1RN:ε2|u|2+K(x)u2<,

endowed with the standard norm

uε,K=ε2|u|2+K(x)u212,

which is induced by the inner product

u,vε,K=ε2uv+K(x)uv.

Now we define H to be the product space Hε,P1×Hε,Q1 with the norm

(u,v)ε2=uε,P2+vε,Q2

and set

Eε,xε=(φ,ψ)H:(φ,ψ),Vε,xεxi,Wε,Xεxiε=0

for i = 1, · · · , N, where

(φ,ψ),Vε,Xεxi,Wε,Xεxiε=ε2φVε,xεxi+P(x)Vε,xεxiφ+ε2ψWε,xεxi+Q(x)Wε,Xεxiψ.

Note that the variational functional corresponding to (1.1) is

(2.1) Iε(v,w)=12ε2|v|2+P(x)v2+ε2|w|2+Q(x)w2μ2v2wγ3w3.

Then IC2(H, ℝ) and its critical points are solutions of (1.1).

Set

Jε(φ,ψ)=IεVε,xε+φ,Wε,xε+ψ,(φ,ψ)Eε,xε.

We can expand Jε(φ, ψ) as follows:

(2.2) Jε(φ,ψ)=Jε(0,0)+ε(φ,ψ)+12Lε(φ,ψ)+Rε(φ,ψ),

where

ε(φ,ψ)=ε2Vε,xεφ+P(x)Vε,xεφ+ε2Wε,xεψ+Q(x)Wε,xεψμVε,xεWε,xεφμ2Vε,xε2ψγWε,xε2ψ=P(x)PxεVε,Xεφ+Q(x)QxεWε,Xεψ,Lε(φ,ψ)=ε2|φ|2+P(x)φ2+ε2|ψ|2+Q(x)ψ2μWε,xεφ2+2γWε,xεψ2+2μVε,xεφψ

and

Rε(φ,ψ)=μ2φ2ψγ3ψ3.

It follows from [5] that Vε,Xε+φ,Wε,xε+ψ is a critical point of Iε(v, w) if and only if (φ, ψ) is a critical point of Jε(φ, ψ). In order to find a critical point for Jε(φ, ψ), we need to discuss each terms in the expansion (2.2).

Lemma 2.1

There exists C > 0 independent of ε such that

Rεi(φ,ψ)CεN2(φ,ψ)ε3i,i=0,1,2,

where Rεi(φ,ψ) denotes the i-th derivative of Rε(φ, ψ).

Proof. Recall that

Rε(φ,ψ)=μ2φ2ψγ3ψ3.

By the direct computations, we get

Rε(φ,ψ),(ξ,η)=μ2φ2η+2φψξγψ2η

and

Rε(φ,ψ)(ξ,η),(g,h)=μ(φgη+φhξ+ψgξ)2γψhη.

Observe that by letting uε(x) = u(εx), it is easy to check

(2.3) uLpRNCε1p12Nuε

with uε=ε2|u|2+u212 for 2p2 and some C > 0. Then we have

RE(φ,ψ)C|φ|2|ψ|+|ψ|3C|φ|323|ψ|313+C|ψ|3CεN2φε2ψε+CεN2φε3CεN2(φ,ψ)ε3,
Rε(φ,ψ),(ξ,η)C|φ|2|η|+|φ||ψ||ξ|+|ψ|2|η|C|φ|323|η|313+C|φ|313|ψ|313|ξ|313+C|ψ|323|η|313CεN2φε2ηε+φεψεξε+ψε2ηεCεN2(φ,ψ)ε2(ξ,η)ε

and similarly,

Rε(φ,ψ)(ξ,η),(g,h)C(|φ||g||η|+|φ||h||ξ|+|ψ||g||ξ|+|ψ||h||η|)CεN2(φ,ψ)ε(ξ,η)ε(g,h)ε.

This completes our proof.

Lemma 2.2

There exists C > 0 independent of ε such that

εCPxε+QxεεN2+1+εN2+2.

Proof. For any (φ,ψ)Eε,xε, taking into account the decay property of U, we find for any fixed d > 0,

ε(φ,ψ)=P(x)PxεVε,xεφ+Q(x)QxεWε,xεψ=εNPεy+xεPxεαPxεUPxεyφεy+xε+Qεy+xεQxεβQxεUQxεyψεy+xε=εNBdε(0)Pεy+xεPxεαPxεUPxεyφεy+xε+Bdε(0)Qεy+xεQxεβQxεUQxεyψεy+χε+RNBdε(0)Pεy+xεPxεαPxεUPxεyφεy+xε
+RNBdε(0)Qεy+xεQxεβQxεUQxεyψεy+xεCεN2Bdε(0)α2Pεy+xεPxε2P2xεU2Pxεy12+Bdε(0)β2Qεy+xεQxε2Q2xεU2Qxεy12(φ,ψ)ε+CεN2RNBdε(0)U2Pxεy12+RNBdε(0)U2Qxεy12(φ,ψ)ε=OεN2eτε(φ,ψ)ε+OεN2dεPxε|εy|+|εy|22U2Pxεy12+εN2Bdε(0)Qxε|εy|+|εy|22U2Qxεy12(φ,ψ)ε=OPxε+QxεεN2+1+εN2+2(φ,ψ)ε

for some τ > 0.

Now it is easy to check that

ε2(vφ+wψ)+P(x)vφ+Q(x)wψμWε,xεvφ+μVε,xεvψ+μVε,xεwφ+2γWε,xεwψ

is a bounded bi-linear functional in Eε,xε. Hence we can define L to be a bounded linear map from Eε,xε to Eε, such that for any (v,w),(φ,ψ)Eε,χε,

L(v,w),(φ,ψ)=ε2(vφ+wψ)+P(x)vφ+Q(x)wψμWε,xεvφ+μVε,xεvψ+μVε,xεwφ+2γWε,xεwψ.

Next we want to discuss the invertibility of L in Eε,Xε. To this end, we denote (V,W) = (αU, βU)). Then (V,W) solves

(2.4) Δv+v=μvw,xRN,Δw+w=μ2v2+γw2,xRN.

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 H1RN×H1RN in the sence that the kernel is given by span λ(μ,γ)Vxi,Wxii=1,2,,N , where λ(μ,γ)0.

Using the above result, we come to discuss the invertibility of L in Eε,Xε.

Lemma 2.4

There exist constants ρ > 0 and ε0 > 0 such that for all ε ∈ (0, ε0],

L(v,w)ρ(v,w)ε,(v,w)Eε,xε.

Proof. We argue by contradiction. Suppose that there exist εn → 0, χεn x0 and (vn , wn) ∈ Eεn,xεn such that for any φn,ψnEεn,xεn,

(2.5) Lvn,wn,φn,ψn=on(1)vn,wnεnφn,ψnεn.

Without loss of generality, we can assume that vn,wnεn2=εnN and let

v˜n(x)=1PxεnvnεnχPxεn+χεn

and

w˜n(x)=1QxεnwnεnxQxεn+χεn.

Then, in view of vn,wnεn2=εnN, we get

v˜n,w˜nH1×H1=v˜n2+v˜n2+w˜n2+w˜n2C,

which implies that up to a subsequence, there exists v, wH1(ℝN) such that as n → +∞,

v˜nv,w˜nw in H1RN,v˜nv,w˜nw in Lloc2RN.

Now we claim that v = w = 0. Considering (2.5), we find

(2.6) P2xεnv˜nφ˜n+w˜nψ˜nμ(βU)v˜nφ˜nμ(αU)w˜nφ˜nμ(αU)v˜nψ˜n2γ(βU)w˜nψ˜n+PxεnPεnyPxεn+xεnv˜nφ˜n+QxεnQεnyQxεn+xεnw˜nψ˜n=PxεnN4on(1)P2xεnφ˜n2+Q2xεnψ˜n2+PxεnPεnyPxεn+xεnφ˜n2+QxεnQεnyPxεn+xεnψ˜n212,

where

φ˜n(x)=1PxεnφnεnXPxεn+xEn,ψ˜n(x)=1QxεnψnεnxQxεn+χεn

and φ˜n,ψ˜nE˜εn,xεn with

E˜εn,xεn=(φ˜,ψ˜):φ˜PxεnxxEnεn,ψ˜QxεnxxεnεnEεn,xεn.

On the other hand, being vn,wnεn2=εnN, we have

(2.7) Pxεnv˜n2+w˜n2+PεnχPxεn+xεnv˜n2+QεnxQxεn+xεnw˜n2=PxεnN21.

So, taking φn=vn,ψn=wn, (2.6) gives

(2.8) Pxεnv˜n2+w˜n2μ(βU)v˜n22μ(αU)v˜nw˜n2γ(βU)w˜n2+PεnxPxεn+xEnv˜n2+QεnxQxεn+xEnw˜n2=on(1)PxεnN21.

Also, since v˜n,w˜nE˜εn,xεn , we obtain

Pxεnv˜n(αU)yi+PεnxPxεn+xεnv˜n(αU)yi+Qxεnw˜n(βU)yi+QεnxQxεn+xεnw˜n(βU)yi=0,

which gives that by letting n → +∞,

(2.9) v(αU)xi+v(x)(αU)xi+w(βU)xi+w(x)(βU)xi=0.

Now we take (φ˜,ψ˜)C0RN×C0RN satisfying

(2.10) φ˜(αU)xi+φ˜(αU)xi+ψ˜(βU)xi+ψ˜(βU)xi=0.

Meanwhile, we can decompose (φ˜,ψ˜)E˜εn,xεn as follows

(φ˜,ψ˜)=φ˜n,ψ˜ni=1Nai,nVεn,λεnεnXPxεn+xεnxi,Wεn,xεnεnXQxεn+xεnxi.

Then from φ˜n,ψ˜nE˜εn,xεn and the definition ofE˜εn,xεn, we get

(2.11) Pxεnφ˜(αU)xi+PεnxPxεn+xεnφ˜(αU)xi+Qxεnψ˜(βU)xi+QεnxQxεn+xεnψ˜(βU)xi+i=1Nai,n(αU)xi(αU)xi+PεnxPxεn+xεni=1Nai,n(αU)xi(αU)xi+i=1Nai,n(βU)xi(βU)xi+QεnxQxεn+xεni=1Nai,n(βU)xi(βU)xi=0.

But by decay property of U, there exists C > 0 such that

(2.12) (αU)xi2+(βU)xi2+(αU)xi2+(βU)xi2C>0.

Hence, combining (2.10)-(2.12), we can easily check that ai,n → 0as n → +∞and then it follows from (2.6) that

(2.13) [P2(x0)(vφ˜+wψ˜+vφ˜+wψ˜μ(βU)vφ˜μ(αU)wφ˜μ(αU)vψ˜2γ(βU)wψ˜)]=0.

This implies that for any (φ,ψ)C0RN×C0RNE˜εn,xεn,

(2.14) [vφ+wψ+vφ+wψμ(βU)vφμ(αU)wφμ(αU)vψ2γ(βU)wψ]=0.

On the other hand, from the fact that (αU, βU) solves (2.4), we see

(2.15) v(αU)xi+w(βU)xi+v(αU)xi+w(βU)xiμ(βU)v(αU)xiμ(αU)w(αU)xiμ(αU)v(βU)xi2γ(βU)w(βU)xi=0,

which, together with (2.14), yields that for any (φ,ψ)C0RN×C0RN

[vφ+wψ+vφ+wψμ(βU)vφμ(αU)wφμ(αU)vψ2γ(βU)wψ]=0.

So (v, w) is a solution of

Δv+v=μ(βU)v+μ(αU)w,xRN,Δw+w=μ(αU)v+2γ(βU)w,xRN.

Using Proposition 2.3, there exist bi ∈ ℝ, i = 1, 2, · · · , N such that

(v,w)=i=1Nbi(αU)xi,(βU)xi.

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 v˜n0 in Lloc2RN and the exponential decay of U, we have

Uv˜n2=BR(0)Uv˜n2+RNBR(0)Uv˜n2=oR(1)+OeR,

where oR(1) → 0 as R → +∞.

As a result, from (2.8) and (2.7), we deduce that

on(1)PxεnN21=Pxεnv˜n2+w˜n2μ(βU)v˜n22μ(αU)v˜nw˜n2γ(βU)w˜n2+PεnxPxεn+xEnv˜n2+QεnxQxεn+xEnw˜n2=PxEnN21+oR(1)+OeR,

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, such that

Jεφε,ψε,(g,h)=0,(g,h)Eε,xε.

Moreover,

φε,ψεεCPxε+QxεεN2+1+εN2+2

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, ε(φ,ψ) is a bounded linear function in Eε,Xε. So using Riesz representation theorem, we obtain that there is an ˉεEε,xε, such that

ε(φ,ψ)=ˉε,(φ,ψ).

Hence finding a critical point for Jε(φ, ψ) is equivalent to solving

(2.16) ε+L(φ,ψ)+Rε(φ,ψ)=0.

It follows from Lemma 2.4 that (2.16) can be rewritten as

(2.17) (φ,ψ)=A(φ,ψ):=L1ˉε+Rε(φ,ψ).

Now we set

Sε=(φ,ψ)Eε,xε,(φ,ψ)εPxε+QxεεN2+1θ+εN2+2θ

with θ ∈ (0, 1). Then for any (φ, ψ) ∈ Sε,

A(φ,ψ)Cˉεε+Rε(φ,ψ)Cˉεε+CεN2(φ,ψ)ε2Pxε+QxεεN2+1θ+εN2+2θ.

Then A maps Sε to Sε. On the other hand, for any (φ1, ψ1), (φ2, ψ2) ∈ Sε, form Lemma 2.1,

Aφ1,ψ1Aφ2,ψ2=L1Rεφ1,ψ1Rεφ2,ψ2CRεtφ1,ψ1+(1t)φ2,ψ2φ1,ψ1φ2,ψ2ε12φ1,ψ1φ2,ψ2ε,

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

φε,ψεεPxε+QxεεN2+1θ+εN2+2θ.

Furthermore, in view of (2.17), we get

φε,ψεε=L1ˉε+L1Rεφε,ψεεCˉεε+CεN2φε,ψεε2Cˉεε+CPxε+QxEε1θ+ε2θφε,ψεε,

which implies that

φε,ψεεCˉεεCPxε+QxεεN2+1+εN2+2.

3 Proof of our main result

In this section, we assume that x0Γ 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,

ε2Vε,y+φε,yg+P(x)Vε,y+φε,yg+ε2Wε,y+ψε,yh+Q(x)Wε,y+ψε,yhμVε,y+φε,ygWε,y+ψε,y+μ2Vε,y+φε,y2h+γWε,y+ψε,y2h=0.

Moreover,

φε,y,ψε,yε=O(|P(y)|+|Q(y)|)εN2+1+εN2+2.

To get a true solution of (1.1), we need to choose y such that

(3.1) ε2vεvεxi+P(x)vεvεxi+ε2wεwεxi+Q(x)wεwεxiμvεvεxiwε+μ2vε2wεxi+γwε2wεxi=0,

where vε=Vε,y+φε,y,wε=Wε,y+ψε,y and i = 1, · · · , N. It is easy to check that (3.1) is equivalent to

(3.2) P(x)xivε2+Q(x)xiwε2=0,i=1,,N,

which is exact the Pohozaev type identity.

For y close to x0, yΓt 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

DζiP(y)=0,|P(y)|=DvP(y)

and

DζiQ(y)=0,|Q(y)|=DvQ(y).

First, we prove the following results.

Lemma 3.2

If (H1)− (H3) hold, then

DvP(x)vε2+DvQ(x)wε2=0

is equivalent to

DvP(y)+DvQ(y)=Oε2.

Proof. By the direct computations, we have

(3.3) DvP(x)Vε,y2+DvQ(x)Wε,y2=2DvP(x)Vε,yφε,y+DvP(x)φε,y2+2DvQ(x)Wε,yψε,y+DvQ(x)ψε,y2=ODVP(y)+DvQ(y)εN2φε,y,ψε,yε+εN2+1φε,y,ψε,yε+φε,y,ψε,yε2=εNODvP(y)+DνQ(y)ε+ε2.

On the other hand, using Taylor’s expansion, we get

(3.4) DvP(x)Vε,y2+DvQ(x)Wε,y2=εNBδε(0)DvP(y)+ε2|X|22!ΔDvP(y)α2P2(y)U2(P(y)x)+Bδε(0)DvQ(y)+ε2|X|22!ΔDvQ(y)β2Q2(y)U2(Q(y)x)+OεN+4=ODvP(y)+DvQ(y)εN+ΔDvP(y)+ΔDvQ(y)εN+2+εN+4.

Combining (3.3) and (3.4), we find

DvP(y)+DvQ(y)=Oε2.

Lemma 3.3

Under the conditions (H1)− (H3),

DζP(x)vε2+DζQ(x)wε2=0

is equivalent to

DςΔP(y)+DςΔQ(y)+B1K(y)ε2=OG1(y)+G2(y)(|P(y)|+|Q(y)|)+G1(y)+G2(y)ε,

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

Gj(x)=i=1NGj(y)yixiyi+12i=1N=1N2Gj(y)yiyxiyixy+o|xy|2, in Bd(y),

we have

G1(x)Vε,y2+G2(x)Wε,y2=2G1(x)Vε,yφε,yG1(x)φε,y22G2(x)Wε,yψε,yG2(x)ψε,y2=OεN2εG1(y)+εG2(y)+ε2φε,y,ψε,yε+εG1(y)+G2(y)φε,y,ψε,yε2=OG1(y)+G2(y)(|P(y)|+|Q(y)|)εN+2+G1(y)+G2(y)εN+3+εN+4.

On the other hand, from Gj(y) = 0, j = 1,2, we get

G1(x)Vε,y2+G2(x)Wε,y2=OεN+2ΔG1(y)+ΔG2(y)+B1K(y)εN+4+εN+6.

As a result, the result follows.

Proof of Theorem 1.4

Now, by Lemmas 3.2 and 3.3, (3.2) is equivalent to

(3.5) DvP(y)+DvQ(y)=Oε2,DςΔP(y)+DςΔQ(y)=O|P(y)|+|Q(y)|+ε2,

which is also equivalent to

(3.6) DvP(y)+DvQ(y)=Oε2,DςΔP(y)+DςΔQ(y)=Oε2.

Let yˉ Γ be the point such that yyˉ=κv for some κ ∈ ℝ. We have DvP(yˉ)=0 and DvQ(yˉ)=0. As a result,

DvP(y)+DvQ(y)=DvP(y)DvP(yˉ)+DvQ(y)DvQ(yˉ)=Dvv2P(yˉ)+Dvv2Q(yˉ)yyˉ,v+O|yyˉ|2,

which, together with the non-degenerate assumption, yields that DνP(y) + DνQ(y) = O(ε2) can be written as

(3.7) yyˉ,v=Oε2+|yyˉ|2.

Let ςˉi be the ith tangential unit vector of Γ at yˉ. It follows from the assumption (H2) that

(DςiΔP)(y)+(DςiΔQ)(y)=(DςˉiΔP)(yˉ)+(DςˉiΔQ)(yˉ)+O(|yyˉ|)=(DςˉiΔP)(yˉ)+(DςˉiΔQ)(yˉ)+O(ε2)

and

(DςˉiΔP)(yˉ)+(DςˉiΔQ)(yˉ)=(DςˉiΔP)(yˉ)(Dςˉi,0ΔP)(x0)+(DςˉiΔQ)(yˉ)(Dςˉi,0ΔQ)(x0)=(TDςi,0ΔP)(x0),yˉx0+(TDςi,0ΔQ)(x0),yˉx0+O(|yˉx0|2),

where ∇T is the tangential gradient on Γ at x0 and ςi,0 is the ith tangential unit vector of Γ at x0. Hence (DςΔP)(y) + (DςΔQ)(y) = O(ε2) can be rewritten as

(3.8) (TDςi,0ΔP)(x0)+(TDςi,0ΔQ)(x0),yˉx0=O(ε2+|yˉx0|2).

So we can solve (3.7) and (3.8) to get y = xε with xεx0 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).

  1. Conflicts of Interest

    Authors state no conflict of interest.

  2. 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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Received: 2021-04-22
Accepted: 2021-07-04
Published Online: 2021-08-25

© 2021 Jing Yang and Ting Zhou, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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