Multi-spikes solutions for a system of coupled elliptic equations with quadratic nonlinearity

. This paper is devoted to study the following systems of coupled elliptic equations with quadratic nonlinearity (cid:40) which arises from second- harmonic generation in quadratic optical media. We assume that the potential functions P ( x ) and Q ( x ) are positive functions and have a strict local maxima at x 0 . Applying the ﬁnite dimensional reduction method, for any integer 1 ≤ k ≤ N + 1, we prove the existence of positive solutions which have k local maximum points that concentrate at x 0 simulta- neously when ε is small.

1. Introduction and main results. In this paper, we study the following system of coupled elliptic equations with quadratic nonlinearity −ε 2 ∆v + P (x)v = µvw, x ∈ R N , −ε 2 ∆w + Q(x)w = µ where ε is a small parameter, P (x) and Q(x) are positive continuous potentials, 2 ≤ N < 6, µ > 0 and µ > γ. System (1) arises from nonlinear optic theory. The cubic nonlinear Schrödinger equation is the basic equation describing the formation and propagation of optical solutions in Kerr-type materials [11,28]. Here ψ 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 ∂ 2 ∂τ 2 for temporal solitons where τ is the normalized retarded time, or ∇ 2 = N i=1 ∂ 2 ∂x 2 i where x = (x 1 , · · ·, x N ) is the direction orthogonal to z. Solitary wave solutions to (2) and its generations have been proved in, for example [7,26].
Experimental physical scientists obtained a powerful source of coherent light which allow them enter a new nonlinear level of optical research when the lasers were invented in 1960s. Second Harmonic Generation (SHG) were discovered when the optical material has a χ (2) (i.e. quadratic) nonlinear response instead of conventional Kerr χ (3) material for which (2) is based on (see [6,8]). Recent progress in materials with high second-order nonlinearities, including polymeric electro-optical waveguide, has stimulated experimental efforts to increase indirectly effective χ (3) nonlinearities taking advantages of cascaded second-order effects. Suppose that we consider a strong parametric interaction of three stationary quasi-plan monochromatic waves with frequencies ω i (i = 1, 2, 3), there is no walk-off between harmonic waves, the frequencies of interacting waves are matched exactly (ω 1 + ω 2 = ω 3 ), and corresponding wave vectors are almost matched k 1 ω 1 + k 2 ω 2 − k 3 ω 3 = ∆k k i where k i = |K i | and wave vectors. Then with some conventional normalization and the assumption that ω 1 = ω 2 = ω3 2 , we can obtain the following system of type-I SHG (see [8]) where v is a renormalized slowly varying complex envelope of wave with frequency ω 1 , w is the one with frequency ω 3 , σ, α > 0, and r, s = ±1. In the spatial solition case r = s = 1, while the temporal case all four combinations for r, s = ±1 are possible. The physically realistic spatial dimensions are N = 1 or N = 2. Then the chirp-free two-wave (symbiotic) solitions can be found as real-valued solutions of the steady state ( ∂ ∂z = 0) equation: In the case of N = 1, the existence of a non-trivial ground state solution of (3) was proved in [33] by using a variational approach. Multi-pulse solutions of (3) for N = 1 were first observed in numerical simulations (see [33]), and the existence of multi-pulse solutions was analytically proved by using singular perturbation theory in [32].
Before presenting our main assumptions and main results, we want to mention that instead of the fewer study for system (4), the following has absorbed a lot of investigation. Fox example the existence of solitary waves to (5) has been explored by many authors in recent years, see [2,3,9,10,13,16,21,22,23,24,27,29,30] and their references therein. Now we are ready to present our main assumptions on P (x), Q(x), we assume that: Without loss of generality, we may assume that P (x 0 ) = Q(x 0 ) = 1. Our main results roughly speaking are as follows. Before giving the more precise statement of our main results, we present some notations first. Let us denote U be the unique ground state of Hereafter, for any function K(x) > 0, we denote ε,Q . Now we assume that (v ε , w ε ) is a solutions of (1) and x ε ∈ R N with x ε → x 0 ∈ R N as ε → 0. We take ( v ε (y), w ε (y)) = (v ε (εy + x ε ), w ε (εy + x ε )).

ZHONGWEI TANG AND HUAFEI XIE
Then one can easily check that ( v ε , w ε ) satisfies So we find that for x near x ε , where (v, w) solves (8). Thus the system (8) in some sense is the limit system of (1). Note that (V, W ) := (αU, βU ) solves (8) provided that µ > γ and We will use (V, W ) := (αU, βU ) as the main component to construct the solutions of (1). We remark that the system (8) possesses a symmetry that if (V, W ) is a solution of (8), so is (−V, W ).
Let k be any positive integer, we denote . Now we are ready to state our main result: Under the assumptions of Theorem 1.1, for any positive integer k ≤ N + 1, there exists ε 0 > 0, such that for each ε ∈ (0, ε 0 ), the problem (1.1) has a solution (v ε , w ε ) of the form where θ 1 , θ 2 are the numbers defined in assumptions (H P ),(H Q ). τ is a positive constant which will be specified in Lemma 2.1.
This paper is organized as follows. In section 2, we will establish some preliminary estimates. In section 3, we will carry out a reduction procedure and then give the proof of our main results. In Appendix, we present some basic estimates for the functional corresponding to problem (1).
2. Some preliminary estimates. In this section, we will present some preliminary estimates which are the main ingredient for the proof of our main results in next section.
Thus by a standard argument (see Cao, Noussair and Yan [12]), one can easily show that (V ε,y + ϕ, W ε,y + ψ) is a critical point of I ε (v, w) if and only if (y, ϕ, ψ) is a critical point of J ε (y, ϕ, ψ). Now we fix y ∈ D ε,δ k first and we want to find a critical point J ε (y, ϕ, ψ) with respect to (ϕ, ψ). In the following of this section, without leading confusion we will omit the dependence of y for the functional J ε and we denote J ε (y, ϕ, ψ) by J ε (ϕ, ψ) instead.
We expand J ε (ϕ, ψ) as follows: In order to find a critical point (ϕ, ψ) ∈ E ε,y for J ε (ϕ, ψ), we need to discuss each term in the expansion. It is easy to check that is bounded bi-linear functional in E ε,y . Thus, there is a bounded linear operator L from E ε,y to E ε,y , such that By the above analysis, we have the following results. First we give the nondegeneracy property of (V, W ).

ZHONGWEI TANG AND HUAFEI XIE
Similarly, we deduce that Hence, we get the conclusion. Now we give an estimate for l ε which is the following lemma.

Lemma 2.3.
There is a constant C > 0 which is independent of ε, such that Proof. By a direct calculation, for any (ϕ, ψ) ∈ E ε,y , we have Similarly, we also have Similarly, we can get Finally, combining the above estimates gives the required estimate.
3. The Finite-Dimensional reduction and proof of the main results. In this section, we intend to prove the main theorem by the Lyapunov-Schmidt reduction. Let L ε be defined as in above section, we have the following estimate.
Then we can obtain ( v n,i , w n,i ) ≤ C.
So we may assume that there exist v, w ∈ H 1 (R N ), such that as n → +∞, v n,i → v, strongly in L 2 loc (R N ), and w n,i → w, strongly in L 2 loc (R N ).
On the other hand, Hence, we can easily check that a j,n,l → 0 as n → ∞ for i = j, while a i,n,l → a i,l up to a subsequence. Taking ( ϕ n , ψ n ) into (10) and letting n → ∞, we obtain that From the fact that (V, W ) solves (8), we see that Since in (11) is arbitrary, the non-degeneracy of (V, W ) yields that there exists b l ∈ R, l = 1, · · ·, N, such that Therefore, (v, w) = (0, 0), which is exactly our claim.
Now we deduce the contradiction as follows.
which is impossible for large n and R. As a result, we complete the proof.
Let us denote y = (y 1 , y 2 , · · · , y k ), we have the following lemma which is the main part in the Liapnouv-Schmidt reduction argument.
We analyze the asymptotic behavior of F (y) with respect to ε. From proposition 2, we have for some constants.
Combining the above estimates yields Now consider the following maximizing problem We claim that y ε is an interior point of D ε,δ k . We will prove the claim by a comparison argument.
Applying the Hölder continuity of P (x) and Q(x), we derive that By using F (y) ≤ F (y ε ), we deduce Then we have which implies that y ε is an interior point of D ε,δ k .
Appendix. In this section, we will give the energy expansion for the approximate solutions.
Proof. Recall that I ε (V ε,y , W ε,y ) = 1 2 R N ε 2 |∇V ε,y | 2 + P (x)V 2 ε,y + ε 2 |∇W ε,y | 2 + Q(x)W 2 Since y j = y i for j = i and V and W decay exponentially at infinity, we get By direct computation, we obtain Similarly, we have Applying the elementary computations, we derive Hence adding the above equalities together, we obtain the desired estimate.