Semiclassical Solutions for a Kind of Coupled Schrödinger Equations

where 2 < p < 2∗, 2 < q < 2∗, N ≥ 3, and 2∗ = 2N/ðN − 2Þ are the Sobolev critical exponent;λ > 0 is a parameter; and a1, a2, a3, b1, b2, b3, c ∈ CðRN ,RÞ and u, v ∈H1ðRNÞ. As it is known in [1], this type of systems arises in nonlinear optics. In the past years, under different kinds of assumptions on the potentialV and the nonlinearity f , many authors [2–8] focus on the following kind of Schrödinger equation:

In [11], the authors investigated standing waves for the following kind of coupled Schrödinger equations: q cannot hold. So in the very recent paper [12], Peng et al. investigated the following coupled Schrödinger equations and generalize the result in [11]: where a 1 , b 1 are the same as in (3), N ≥ 3. Under the following conditions, (A1) a 1 ðxÞ ≥ a 1 ð0Þ = 0 and b 1 ðxÞ ≥ 0, and there exist constants a 0 1 > 0 and b 0 1 > 0 such that the measure of the sets A a 0 1 ≔ fx : a 1 ðxÞ < a 0 1 g and B b 0 1 ≔ fx : b 1 ðxÞ < b 0 1 g are finite (A2) there exists a constant ϑ ∈ ð0, 1Þ such that jcðxÞj 2 ≤ ϑa 1 ðxÞb 1 ðxÞ for all x ∈ ℝ N ; Peng et al. proved that system (4) has at least one nontrivial solution. An interesting question is what will happen if the nonlinearity is also critical growth in system (4)? Motivated mainly by the abovementioned results, we will answer this question and prove that system (1), under conditions (A1) and (A2), and (A3) there exist constants a 0 2 , possesses nontrivial solutions if λ ∈ ð0, λ 0 Þ, where λ 0 is related to a 1 , a 2 , a 3 , b 1 , b 2 , b 3 , and N. As far as we know, similar results for system (1) with a critical exponent have not been investigated by variational methods in the literature.
The following condition is similar to condition (A1): (A1') b 1 ðxÞ ≥ b 1 ð0Þ = 0 and a 1 ðxÞ ≥ 0, and there exist constants a 0 1 > 0 and b 0 1 > 0 such that the measure of the sets Since η 0 and η 2 * are embedding constants and ω N is the volume of the unit ball in ℝ N . From (A1') and (A1), using b 1 ð0Þ = 0 and a 1 ð0Þ = 0, one can let μ 0 > 1 such that Let w = ðu, vÞ and λ −2 = μ, then system (1) can be rewritten as and the functional of (9) is given by As is known, the solutions of (1) are the critical points of S λ −1/2 ðwÞ. The main results are the following.
Remark 3. Since the presence of the terms a 2 ðxÞjuj p−2 u, a 3 ðxÞjuj 2 * −2 u, b 2 ðxÞjvj p−2 v, and b 3 ðxÞjvj 2 * −2 v, system (1) is more general than (4), and it is more difficult to deal with the nontrivial solutions. In order to prove that system (1) has nontrivial solutions, we need to find some conditions to restrict a 2 ðxÞ, a 3 ðxÞ, b 2 ðxÞ, and b 3 ðxÞ. It seems that there is no literature considering system (1).

Preliminaries
Let Advances in Mathematical Physics From Lemma 1 of [17], by (A1) or (A1') and the Sobolev inequality, there exists a positive constant η 0 > 0 independent of μ such that where H 1 ≔ H 1 ðℝ N Þ. Then, ðE, k·k μ † Þ is a Banach space for μ ≥ 1 equipped with the norm given by (12). Moreover, for s ∈ ½2, 2 * , one has where kwk s is the usual norm in space L s ðℝ N Þ. From (12), we rewrite S μ as It is not difficult to see that S μ ∈ C 1 ðE, ℝÞ and As in [12,22], let Then, θ ∈ H 1 ðℝ N Þ; moreover, In the next section, we will prove the main results.

Proof of the Main Results
Proof of Theorem 1. The proof of Theorem 1 is divided into four steps.

Advances in Mathematical Physics
Step 2. Let c * μ = min fS μ ðte μ , 0Þ, S μ ð0, te μ Þg, we should prove that there exists a constant c μ ∈ ð0, c * μ and a sequence fw n g ⊂ E satisfying By a standard argument, one can obtain (23) by employing the mountain-pass lemma without the (PS) condition, so we omit the details here.