The existence of positive solutions for the Neumann problem of p-Laplacian elliptic systems with Sobolev critical exponent

The paper aims to consider a class of p-Laplacian elliptic systems with a double Sobolev critical exponent. We obtain the existence result of the above problem under the Neumann boundary for some suitable range of the parameters in the systems.

When p = 2, the Dirichlet boundary problems for the system (1.1) are generally studied. Due to the "loss of compactness, " a direct application of the classical method is invalid for such problems. In [1], Brezis and Nirenberg first studied the Dirichlet problem with a critical Sobolev exponent for the following nonlinear elliptic equation: (1. 2) In order to prove the existence of a sequence satisfying the (PS) (C) condition, the authors used a function which can achieve the best Sobolev constant as the test function to esti-mate the energy. This overcame the difficulties which were caused by the embedding with the critical exponent that made the minimization sequences not compact. In recent years, the problem of elliptic equations (systems) with critical exponents has attracted the attention of many researchers, and related theory has also made great progress, e.g., results on the existence and nonexistence of nontrivial and multiple solutions were proved, and also some properties of solutions (see [2][3][4][5][6][7][8]). At the same time, in recent years, the problem with critical exponents in fractional elliptic equations, such as fractional Kirchhoff problems and Schrödinger-Kirchhoff-type problems involving the fractional p-Laplacian, have also attracted the attention of researchers (see [9][10][11]). The nonlinear elliptic system (1.1) arises from mathematical physics, when studying Bose-Einstein condensation, some reaction-diffusion shadow systems, or static problems of chemotaxis models, and any more. It also has a wide range of applications in mathematical biology, financial mathematics, and so on; more applications can be found in [12][13][14][15][16]. When p = 2, α = β = 0, equation (1.1) has the following form: This problem has attracted the attention of many researchers, and many interesting and important results have been proved [17][18][19]. In [18], by using the minimax theorem and mountain pass lemma, Wang proved that (1.3) has a nontrivial solution when the minimax is lower than the threshold value 1 2N S N 2 and λ > 0 is appropriately large. In [17], the existence of the least energy solutions u λ is proved, under some conditions, i.e., ∃λ 0 > 0 such that, when λ > λ 0 , Q(u λ ) = S λ and When p = 2, α, β > 0, Yang (see [20]) used the method of the blow-up to solve the single critical growth problem as follows: 4) and the author also discussed the asymptotic behavior of the least energy solution when μ → ∞ and λ → ∞. The concentration-compactness principles are very different when comparing Dirichlet and Neumann boundary problems. In [21], Chabrowski and Yang used the concentration-compactness principle under the Neumann boundary to study the existence of the least energy solution of the above equations with a potential form, and also discussed the concentration phenomenon.
With the in-depth study of problem (1.3), researchers have extended this kind of problem to more general p-Laplacian equations or systems. Because of the wide and practical application background, researchers have used the equations in pharmacology, biology, non-Newtonian fluids, etc., and got crucial results [22][23][24][25]. In [26], Wang discussed the following equation with Neumann boundary and critical exponent and proved that it has positive solutions: For the p-Laplacian equation with Neumann boundary and subcritical growth exponents, we can use a standard variational principle to deduce the existence results for the solutions. However, for the problem (1.5), the corresponding energy functional loses compactness, so the standard variational principles are invalid. The Dirichlet boundary problems corresponding to such equations have also been widely studied (see [1,27,28]). However, the method of dealing with the Dirichlet boundary problem is no longer applicable to Neumann boundary problem owning to the best Sobolev constant of the embedding W 1,p 0 ( ) → L P * ( ), which is only depends on N, p, and not on . Under the following conditions, Wang proved that (1.5) has a positive solution by virtue of the local convexity at a point of ∂ : However, the problem with a doubly critical growth involved in this paper is rarely studied. When p = 2, in [29], Peng, Peng, and Wang considered the doubly critical Dirichlet boundary problem: (1.6) In [30], the authors considered the single equation case of the above problem, for the embedding D 1,2 (R N ) → L 2 * (R N ), and they deduced that the following radial function U(x) achieves the best Sobolev constant: With similar methods, the authors of [29] proved that the following form of the best Sobolev constant can be achieved: The authors derived the uniqueness result for the least energy solutions when α, β, N satisfied some conditions, and the form of the least energy solution is (sU x 0 ,ε , bU x 0 ,ε ). We are interested in whether we can use this form of extremal function to research the Neumann boundary problems for the p-Laplacian system whose positive solutions exist in more general case. Therefore, we select the extremal function of the above-mentioned form and use a similar method to that in [1], choose an extremal function as the test function for the energy estimation as in [26], and also need to make the energy corresponding to the Palais-Smale sequence lower than the threshold. Subsequently, we overcome the difficulties caused by the appearance of doubly critical terms |u| p * , |v| p * , |u| α |v| β in (1.1), which leads to the emergence of noncompactness, and, naturally, the existence of a positive solution for the equations (1.1) can be obtained. Now, the main theorem of this paper can be presented as follows: Theorem 1.1 If the following conditions for the parameters α, β, N are satisfied: then there is at least one positive solution for (1.1), when λ 1 , λ 2 are sufficiently large.

Preliminary results
In this section, it is necessary to present some definitions and preliminary lemmas, which are going to be used to prove our basic estimates and main results. First, denote X = W 1,p ( ) × W 1,p ( ), where its norm is as follows: Here W 1,p ( ) is a Sobolev space, and it has the following norm: We define its corresponding energy functional J : X → R as follows: Now, we define the weak solution: In order to prove that the least energy solution of (1.1) exists, and derive the solution's form, we consider the following equation: From [30], we know that, when = R N , the function is a radial function which solves (2.1). Meanwhile, the best Sobolev constant S can be achieved by U(x) for the embedding From [30], we know that the (N + 2)-dimensional manifold of the following form consists of almost all functions which can achieve the best Sobolev constant S: (2.5) Supposing (sU x 0 ,ε , bU x 0 ,ε ) is a positive solution corresponding to the problem (1.1), we have Therefore, and Subsequently, we find that if all of the least energy solutions of (1.1) have the form (sU x 0 ,ε , bU x 0 ,ε ), where s, b are constants, then we know By a similar method as that in [29], under the assumptions of Theorem 1.1, we derive that the following form is suitable for the least energy solutions of (1.1), which can be found in [29]: , then there is a constant C(δ), which depends on δ, such that the following conclusions hold: (1) When h ≡ 0, (2) By a translation transformation, letting y = x , y n = x nh(x ) > 0, we straighten the boundary of D, and complete the proof.
First of all, we have the following result which is necessary to verify that the equations (1.1) satisfy the mountain pass lemma: Lemma 2.3 When the assumptions of Theorem 1.1 are satisfied, the following conclusions hold: ( Proof (1) Because W 1,p ( ) → L p * ( ) is continuous, by Hölder inequality, we have: (2.10) Hence, As a consequence of 1 p -1 p * = 1 N > 0, choosing (u, v) = r small enough, we obtain the desired result J(u, v) ≥ δ > 0.
Next, we introduce the following lemmas (see [31]), which are useful in verifying that the energy functional is lower than the threshold, and whose functional corresponds to the Palais-Smale sequence.

Lemma 2.4
If {u n } is a bounded sequence in L p ( ) such that u n → u a.e., then lim n→∞ |u n | p dx -|u n -u| p dx = |u| p dx. (2.11) (2.12) Lemma 2.6 (see [32]) Assume {u n } is the (PS) sequence corresponding to J λ,p , and u n u, then it has finitely many points in , we denote them by x 1 , x 2 , . . . , x k ∈ , which make u n → u hold in W  (2.14) Then the system (1.1) has a solution (u, v) ∈ X and J(u, v) ≤ c.
As a consequence, we can see that (2.16) Setting (ϕ, ψ) = (u n , v n ) and substituting it into equation (2.16), we then have (2.17) Combining (2.17) with (2.15), we obtain This implies that (u n , v n ) is bounded, hence, ∃C such that (u n , v n ) ≤ C, u n W 1,p ( ) ≤ C, and v n W 1,p ( ) ≤ C. Moreover, there exist u ∈ W 1,p ( ), v ∈ W 1,p ( ) such that (u n , v n ) (u, v) in X. By Lemma 2.6, we have the following results: By calculating the limit of both sides of (2.16), we obtain (2.18) By Definition 2.1, we know that (u, v) is a weak solution of (1.1). We need to verify that (u, v) is a nontrivial solution in the following. Let ω n = u nu, σ n = v nv, then by Lemmas 2.4, 2.5, and 2.6, we see that By (2.15) and the fact J(u n , v n ) → c, J (u n , v n ) → 0, where (u n , v n ) ⊂ X, we have Hence, For ∀ε > 0, where ε is a small suitable positive constant, we denote by (φ i ) m i=1 a partition of unity on¯ , satisfying ∀i, diam(supp φ i ) ≤ ρ, where diam(D) means the diameter of the domain D. Therefore, by Lemma 2.2, provided that ρ is small enough, we have Employing Young inequality with ε, for ∀1 ≤ i ≤ m, u, v ∈ W 1,p ( ), we have )l p p * , and we discuss the following two cases: If l = 0, then it is easy to see that (u n This contradicts c < min{ 1 N S If l = 0, that is, l ≥ 1 2 S N p α,β , then we only need to verify u ≡ 0, v ≡ 0. (i) Assume one of u, v equals zero. It is natural to suppose u ≡ 0, v ≡ 0.
From J (u n , v n ), (u n , v n ) → 0, it is easy to obtain Due to (2.19) and l ≥ 1 2 S N p α,β , we see Similarly, this is a contradiction to c < min{ 1 N S N p α,β , which is also a contradiction. In summary, (u, v) is a nontrivial solution of system (1.1).
By similar calculations as for K 1 (ε), we obtain (3.14) Similarly as for (3.11), As before, performing a spherical coordinate transformation, we get Denoting Estimating II 2 ε gives and so we have proved (3.8).
On the other hand, Finally, we will complete the proof of Theorem 1.1.