Positive solution for an elliptic system with critical exponent and logarithmic terms: the higher dimensional cases

In this paper, we consider the coupled elliptic system with critical exponent and logarithmic terms:

To study the solitary wave solutions of the system (1.1), we set Ψ 1 (x, t) = e ıλ1t u(x) and Ψ 2 (x, t) = e ıλ2t v(x), then system (1.1) is reduced to the following coupled elliptic system with logarithmic terms      x ∈ ∂Ω. ( The logarithmic terms present in themselves many mathematical interests and difficulties.It is easy to see that u = o(u log u 2 ) for u very close to 0. Compared to the critical term |u| 2p−2 u, the logarithmic term u log u 2 has a lower-order term at infinity.Additionally, the logarithmic term has indefinite sign and makes the structure of the corresponding functional complicated.Problem (1.2) is connected to the following Bose-Einstein condensates coming from the Gross-Pitaevskii coupled equations.
Starting from the cerebrated work by Brézis and Nirenberg [5], this critical system has received great attention in the past thirty years, in particular for the existence of positive least energy solutions, we refer to [7,8,16,22] and references therein.
For the single equation setting of (1.2), Deng et al. [12] investigated the existence of positive least energy solutions for the following related single equation In [12], the authors proved that equation (1.3) has a positive least energy solution if λ ∈ R, θ > 0 for all N ≥ 4. Also, they obtained some existence and nonexistence results under other conditions, we refer the readers to [12] for details.Recently, the authors studied the particular case p = 2 and N = 4 in [13], and proved various existence and nonexistence results for the system (1.2).
In this paper, we continue our previous work [13] to study the existence and nonexistence of positive solutions for system (1.2) in the higher dimensional case.That is, we always work under the assumptions N ≥ 5 and 2p = 2 * .
The higher dimensional case introduces different phenomena and challenges compared to the specific case where N = 4, since the critical exponent is given by 2p ∈ (2, 4) for N ≥ 5, whereas for N = 4, it is 2p = 4.This brings some difficulties and requires us to develop some new ideas.
The reason is that p = 2 if N = 4, whereas 1 < p < 2 if N ≥ 5.The fact that 1 < p < 2 introduces significant differences for β > 0, which means that the method used in [13] cannot apply directly to this paper, and so we require some new ideas and techniques.As we will see in Proposition 2.4, the approach to establish energy estimate for β > 0 is completely different from that in the case N = 4 [13].
(2) In the particular case θ = 0, [8, Theorem 1.3] said that the system (1.2) has a positive least energy solution for any β ̸ = 0 and λ 1 , λ 2 ∈ (0, λ 1 (Ω)).Comparing this with Theorem 1.1, the logarithmic term θ i u log u 2 has a significant impact on the existence of solutions, and introduces different and more challenging situations.One of the main difficulties comes from the uncertain sign of the logarithmic term, which implies that the Nehari set N may not be a C 1 -manifold for all β < 0. As a result, N cannot be a natural constraint.So we have to restrict the range of β to β ∈ (−β 0 , 0) and demonstrate that we can find the free critical points on a special set, see Proposition 2.1 and 2.2.Another difficulty is do to the presence of logarithmic terms, which makes the structure of functional complicated by using variational method.This requires us to make more careful calculations and develop new ideas and innovative techniques when establishing the energy estimates, we refer to Step 1 and 2 in the proof of Proposition 2.4 for details.Now we focus on the existence when θ 1 , θ 2 < 0 and consider the following special sets, Here, and S denotes the Sobolev best constant of The following two existence results Theorem 1.2 and 1.3 correspond to Theorem 1.2 and 1.3 in [13] (which dealt with the critical case N = 4).
where B r := {(u, v) ∈ H : |∇u| 2 2 + |∇v| 2 2 < r} and ρ will be given by Lemma 3.1.When β < 0, we assume that one of the following holds: ; when β > 0, we assume that there exists ϵ > 0 such that one of the following holds: (iv 2) has a positive solution (ũ, ṽ) such that L(ũ, ṽ) = C ρ < 0, which is a local minima.Remark 1.2.Comparing with the case θ 1 , θ 2 > 0, the presence of negative θ 1 and θ 2 changes the geometry of L, and makes it not straightforward to obtain the Palais-Smale sequence in N .The assumptions (i)-(vi) establish the existence of a local minimum C ρ for the functional L, thus we can obtain the corresponding Palais-Smale sequence by taking minimizing sequence for C ρ , and we prove that the weak limit of Palais-Smale sequence is a positive solution of system (1.2).Unfortunately, we do not know whether it is a positive least energy solution. where Assume that the conditions stated in Lemma 3.1 hold, and Then the system (1.2) possesses a nonnegative solution (û, v) ̸ = (0, 0) such that L(û, v) = C K < 0.
Remark 1.3.In Theorem 1.3, we obtain a nonnegative solution (û, v).However, we do not know whether it is a positive least energy solution since we cannot prove that (û, v) is fully nontrivial.
For the nonexistence of positive solutions for the system (1.2), we have the following results. where If β > 0, then the system (1.2) has no positive solutions.
Remark 1.4.In [12], the authors obtained the existence of positive solution for (1.4)only in the case N = 3, 4 and θ < 0 under certain additional conditions.But they do not give the existence result of positive solutions for the general case N ≥ 5 and θ < 0. Here, we give a positive answer to this question in Theorem 1.5, 1.6, and improve the results in [12].Furthermore, we give the type of the positive solution (a local minimum or a least energy solution) and show that its energy level is negative.
Before closing the introduction, we give the outline of our paper and introduce some notations.In Section 2, we will prove Theorem 1.1.In Section 3, we will prove Theorem 1.2, 1.3 and 1.4.In Sections 4, we will prove Theorem 1.5 and 1.6.
Throughout this paper, we denote the norm of L p (Ω) by | • | p for 1 ≤ p ≤ ∞.We use "→" and "⇀" to denote the strong convergence and weak convergence in corresponding space respectively.The capital letter C will appear as a constant which may vary from line to line, and C 1 , C 2 , C 3 are prescribed constants.

Proof of Theorem 1.1
In this section, we always assume that N ≥ 5 and (λ i , µ i , θ i ) ∈ Σ 1 for i = 1, 2. Now we establish both lower and upper uniform estimates on the L 2p -norms of elements in the Nehari set that fall below a certain energy level.
, there holds Here, Proof.Take any Since we have the inequality then by the Sobolev inequality and the fact that β < 0, we have On the other hand, we have (2.1) Moreover, Therefore, we have Recalling the following useful inequality (see [19] or [15,Theorem 8.14]) for any τ ∈ (0, 1), we have ) where we fix a > 0 with a π θ 1 < 1 2 and a π θ 2 < 1 2 .Therefore, combining this with (2.3), we can see that there exists Similarly, we can prove that |v p , and the set Then we show that the set N ∩ Q is a natural constraint when β < 0.
Proposition 2.1.Assume that β < 0 and the energy level Take (u, v) ∈ N ∩ Q.Since the matrix M (u, v) is strictly diagonally dominant and β < 0, we have Similarly, we have For that purpose, we only need to show that , where (w, z) = (t 1 u, t 2 v).Therefore, the claim is true.Now we suppose that C N is achieved by (u, v) ∈ N ∩ Q.By the Sobolev embedding theorem, the set N ∩Q is an open set of N in the topology of H. Thus (u, v) is an inner critical point of L in an open subset of N , and in particular it is a constrained critical point of L on N .Since the set N is a C 1 -manifold of codimension 2 in H in a neighborhood of (u, v) ∈ N ∩ Q.Then by the Lagrange multipliers rule, there exist Multiplying the above equation with (u, 0) and (0, v), we deduce from Since the system (2.5)-(2.6)has a strictly positive determinant, by the Cramer's rule, the system has a unique solution Let where C 1 , C 2 are given by Lemma 2.1.Then the following result, together with Proposition 2.1, show that the are in fact the free critical points of L. .
Consider the Brézis-Nirenberg problem with logarithmic perturbation where As in [12], we define the associated modified energy functional and the level where Then we have the following proposition, which plays crucial role in the proof of Theorem 1.1. .
Proof.The main idea of the proof is similar to the proof of [13,Proposition 2.4] in the case N = 4, here we give the details for the sake of clarity and completeness.Without loss of generality, we prove that .
By [12], the energy level C θ1 can be achieved by a positive solution u θ1 .Moreover, we know that u θ1 ∈ C 2 (Ω) and u θ1 ≡ 0 on ∂Ω.Then, there exists a ball .
Then by [5] or [21, Lemma 1.46], we obtain the following (2.8) Also, by [12, Lemma 3.4], we have the following Moreover, one has (2.10) Then by (2.7), we infer that Now we claim that there exists s 1,ε , s 2,ε > 0 such that For that purpose, we consider For ε small enough, we can see from (2.8) and (2.11) that the matrix is strictly diagonally dominant, so it is positive definite.Therefore, there exists a constant C > 0 such that Then It follows from 2p ≥ 2 and lim s→+∞ s 2p . This implies that there exists a global maximum point Without loss of generality, we assume that s 1,ε = 0 and s 2,ε ̸ = 0.
Combining this with (2.13) and (2.16), we have Similarly, we can also prove that . By [12, Lemma 3.3], we can easily see that 2 for i = 1, 2. Therefore, we have .
The proof is completed.
Proof of Theorem 1.1 for the case β < 0. Repeating the proof of Theorem 1.3 for the case β < 0 in [8] with some slight modifications, we can construct a Palais-Smale sequence at the level C N .Then there exists a sequence for n large enough.Then by Lemma 2.1 and (2.4), we can see that {(u n , v n )} is bounded in H. Hence, we may assume that Passing to subsequence, we may also assume that By using the inequality |s 2 log s 2 | ≤ Cs 2−τ + Cs 2+τ , τ ∈ (0, 1) and the dominated convergence theorem, one gets Then by (2.17), we have L ′ (u, v) = 0.Moreover, by using the weak-lower semicontinuity of norm, we have (2.18) Then by the Brézis-Lieb Lemma (see [6] and [8,Lemma 3.3]).we have (2. 19) By a direct calculation, one gets Passing to subsequence, we may assume that Letting n → +∞ in (2.21), we have Now we claim that u ̸ ≡ 0 and v ̸ ≡ 0.
Case 1. u ≡ 0 and v ≡ 0. Firstly, we prove that k 1 > 0 and k 2 > 0. Without loss of generality, we assume by contradiction that k 1 = 0, then we can see that w n → 0 strongly in H 1 0 (Ω) and u n → 0 strongly in H 1 0 (Ω), then by the Sobolev inequality, we can see that u n → 0 strongly in L 2p (Ω), which is impossible by Lemma 2.1.Therefore, we have that k 1 > 0 and k 2 > 0. Since (2.20) holds, it follows from the proof of [8,Theorem 1.3] that there exists t n , s n > 0 such that (t n w n , s n z n ) ∈ N , which is given by (2.26).Moreover, Therefore, by (2.27) we have a contradiction with Proposition 2.3.Therefore, Case 1 is impossible.
Without loss of generality, we may assume that u ≡ 0, v ̸ ≡ 0. Then by Case 1, we have that k 1 > 0, and we may assume that k 2 = 0. Then we know that |w . By (2.20), we have Notice that v is a solution of −∆w = λ 2 w + µ 2 |w| 2 w + θ 2 w log w 2 , we have L(0, v) ≥ C θ2 .Therefore, by (2.22) we have that which is a contradiction with Proposition 2.3.Therefore, Case 2 is impossible.Since Case 1 and 2 are both impossible, we get that u ̸ ≡ 0 and v ̸ ≡ 0. Therefore, (u, v) ∈ N and by (2.22) we have that L(u, v) = C N .Then combining (2.18) with Proposition 2.1 and 2.2, (u, v) is a solution of system (1.2).Since L ′ (u, v) = 0, we can see that which implies that u ≥ 0, v ≥ 0. By the Morse's iteration, the solutions u, v belong to L ∞ (Ω).Then the Hölder estimate implies that u, v ∈ C 0,γ (Ω) for any 0 < γ < 1.
Then we follow the arguments in [12,20], and get that u, v ∈ C 2 (Ω) and u, v > 0 in Ω.This completes the proof.
It remains to prove the Theorem 1.1 for the case β > 0. For that purpose, we first introduce some definitions and lemmas.From now on, we assume that β > 0. Let
On the other hand, let φ 1 , φ 2 ∈ H 1 0 (Ω) \ {0} be fixed positive functions, then for any t > 0, we have which is guaranteed by 2 ≤ 2p < 4 and lim t→+∞ t 2p t 2 log t 2 = +∞.Therefore, we can choose t 0 > 0 large enough such that L(t 0 φ 1 , t 0 φ 2 ) < 0, and This completes the proof.Now we consider the following limit system (2.25) We consider the level From [8, Theorem 1.6], we know that when β < 0, (2.27) Proposition 2.4.For any β > 0, we have Proof.The proof is inspired by [8,Lemma 3.4], but the logarithmic terms in (1.2) make the proof much more delicate, and we require some new ideas.To prove this proposition, we divide the proof into two steps.
Step 1.We prove that B < A. By [8, Theorem 1.6], A can be achieved by a positive least energy solution (U, V ) of the system (2.25), which is radially symmetric decreasing.Moreover, there exists a constant C > 0, such that Without loss of generality, we assume that 0 ∈ Ω.Then there exist a ball B 2R (0 It follows from [8,Lemma 3.4] and the choice of ξ that Moreover, we claim that In fact, by a similar argument as used in that of [8, Lemma 3.4], we have Note that and Therefore, we have Similarly, we can prove that Because of the presence of logarithmic terms in system (1.2), we also need the following new inequalities, Since |s 2 log s 2 | ≤ C for 0 ≤ s ≤ 1, we have and Note that By (2.28), there exists R 0 > 0 such that then we have it is easy to see that there exists a constant C > 0 such that B R 0 (0) |U 2 log U 2 | ≤ C. On the other hand, since f (s) = |s log s| is increasing for 0 ≤ s ≤ e −1 , we can see from (2.32) that Similarly, we can prove that Let h(t) := L(tu ε , tv ε ).By Lemma 2.2, h(0) = 0 and lim t→+∞ h(t) = −∞, we can find t ε ∈ (0, ∞), such that which is equivalent to the following (2.33) Recall that E(U, V ) = A, we have Combining this with (2.29) and (2.31), we deduce from (2.33) that, as which implies that there exists c 1 > 0 such that t ε < c 1 for ε small enough.
Proof.Since (u, v) ∈ M and s log s ≤ (p − 1) −1 e −1 s p for any s > 0, we have Therefore, there exists a constant C 3 > 0 such that Ω (|u This completes the proof. Proof of Theorem 1.1 for the case β > 0. Assume that β > 0. By Lemma 2.2 and the mountain pass theorem (see [2,21]), there exists a sequence Lemma 3.2.Assume that the conditions stated in Lemma 3.1 hold, then we have −∞ < C ρ < 0, where C ρ is given by Theorem 1.2.
. Then for large n, we have Since θ 1 < 0, using the inequality t log t ≥ −e −1 , one gets Similarly, we have since θ 2 < 0. For n large enough, we have Then there holds −∞ < C K < 0.
Proof.Notice that (u, v) ∈ K where (u, v) is the solution given by Theorem 1.2.Hence C K ≤ C ρ < 0. Now we show that C K > −∞.For any (u, v) ∈ K we have Then we deduce from 2p > 2 that C K > −∞.
Proof of Theorem 1.2 and 1.3.Applying Lemma 3.1-3.4,the proofs of Theorem 1.2 and 1.3 follow exactly the same steps as the proofs of Theorem 1.2 and 1.3 in [13], respectively.

The Brézis-Nirenberg problem with logarithmic perturbation
In this section, we prove Theorem 1.5 and 1.6.Firstly, we have the following basic fact.
Proof.The proof can be found in [12], so we omit it.
Proof of Theorem 1.5.Similar to Lemma 3.2 we obtain that −∞ < C ρ < 0. By Lemma 4.1, we can take a minimizing sequence {u n } for C ρ with |∇u n | 2 < ρ − τ and τ > 0 small enough.By Ekeland's variational principle, we can assume that J ′ (u n ) → 0. Similar to Lemma 3.3, we can see that {u n } is bounded in H 1 0 (Ω).Hence, we may assume that u n ⇀ u weakly in H 1 0 (Ω).Passing to subsequence, we may also assume that u n ⇀ u weakly in L 2p (Ω), u n → u strongly in L q (Ω) for 2 ≤ q < 2p, u n → u almost everywhere in Ω.