Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 52, pp. 1–24. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF POSITIVE SOLUTIONS FOR BREZIS-NIRENBERG TYPE PROBLEMS INVOLVING AN INVERSE OPERATOR PABLO ÁLVAREZ-CAUDEVILLA, EDUARDO COLORADO, ALEJANDRO ORTEGA Abstract. This article concerns the existence of positive solutions for the second order equation involving a nonlocal term −∆u = γ(−∆)−1u+ |u|p−1u, under Dirichlet boundary conditions. We prove the existence of positive solu- tions depending on the positive real parameter γ > 0, and up to the critical value of the exponent p, i.e. when 1 < p ≤ 2∗ − 1, where 2∗ = 2N N−2 is the critical Sobolev exponent. For p = 2∗ − 1, this leads us to a Brezis-Nirenberg type problem, cf. [5], but, in our particular case, the linear term is a nonlo- cal term. The effect that this nonlocal term has on the equation changes the dimensions for which the classical technique based on the minimizers of the Sobolev constant, that ensures the existence of positive solution, going from dimensions N ≥ 4 in the classical Brezis-Nirenberg problem, to dimensions N ≥ 7 for this nonlocal problem. 1. Introduction In this work, we analyze the existence of positive solutions of the second order elliptic equation under homogeneous Dirichlet boundary conditions and involving a non-local term, −∆u = γ(−∆)−1u+ |u|p−1u in Ω, u = 0 on ∂Ω, (1.1) where γ is a positive real parameter and Ω is a smooth bounded domain of RN , with N ≥ 3, 1 < p ≤ 2∗ − 1, where 2∗ = 2N N−2 is the critical Sobolev exponent. For p = 2∗ − 1, this problem is critical and there is an important issue with the dimension N . Indeed, we will ascertain the existence of positive solutions at the critical exponent for dimensions N ≥ 7, following similar arguments to those used by Brezis-Nirenberg [5]. Let us first observe that, at the critical exponent p = 2∗−1, problem (1.1) can be seen as a linear perturbation of the critical problem −∆u = |u|2 ∗−2u in Ω ⊂ RN , u = 0 on ∂Ω. (1.2) 2010 Mathematics Subject Classification. 35G20, 35A15, 35J50, 35B38, 35J91. Key words and phrases. Critical point; concentration compactness principle; mountain pass theorem. c©2021. This work is licensed under a CC BY 4.0 license. Submitted November 3, 2020. Published June 14, 2021. 1 2 P. ÁLVAREZ-CAUDEVILLA, E. COLORADO, A. ORTEGA EJDE-2021/52 for which, after applying the well-known result by Pohozaev [11], one can prove the non-existence of positive solutions under the star-shapeness assumption on the domain Ω. Moreover, the classical Brezis-Nirenberg problem −∆u = γu+ |u|2 ∗−2u in Ω ⊂ RN , u = 0 on ∂Ω. (1.3) can be seen as well as a linear perturbation of problem (1.2). In their pioneering paper [5] it was proved that, for N ≥ 4, there exists a positive solution of (1.3) if and only if the parameter γ belongs to the interval (0, λ1), being λ1 the first eigenvalue for the Laplacian operator under homogeneous Dirichlet boundary conditions. Note that, in our situation, the non-local term γ(−∆)−1u plays actually the role of γu in (1.3). Here, we just arrive at the existence of positive solutions for N ≥ 7. Our main motivation to study (1.1) comes from the fourth-order equation, under homogeneous Navier boundary conditions, (−∆)2u = γu+ (−∆)|u|p−1u in Ω ⊂ RN , u = 0 = ∆u on ∂Ω. (1.4) Thus, positive solutions of (1.4) can be seen as positive steady-state solutions of the fourth-order parabolic Cahn–Hilliard type equation, ∂u ∂t + (−∆)2u = γu+ (−∆)|u|p−1u, in Ω× R+, assuming bounded smooth initial data u(x, 0) = u0(x). The latter equation has been previously studied in [2, 1] either for bounded domains or the whole RN , where the authors proved existence and multiplicity of solutions in the subcritical range 1 < p < 2∗ − 1. However, those solutions could change sign. In this work we study the existence of positive solutions for both subcritical case, and the critical one with p = 2∗−1 which creates many more difficulties to deal with than the subcritical case. Let us recall that, thanks to the Sobolev’s Embedding Theorem, we have the compact embedding H1 0 (Ω) ↪→ Lp+1(Ω), (1.5) for 1 ≤ p < 2∗ − 1, being a continuous embedding up to the critical exponent p = 2∗ − 1. Thus, there exists a positive constant C := C(N, p) such that ‖u‖Lp+1(Ω) ≤ C‖u‖H1 0 (Ω), (1.6) for any u ∈ H1 0 (Ω) and 1 ≤ p ≤ 2∗ − 1. Note that here, for the fourth-order elliptic problem (1.4), the Sobolev’s critical exponent is 2∗ = 2N N−2 , since this operator has the representation, (−∆)2u− (−∆)|u|p−1u = (−∆)((−∆)u− |u|p−1u), so that, the necessary embedding features are governed by a standard second-order equation, −∆u = |u|p−1u, justifying the choice for our critical exponent and the relation between (1.1) and (1.4). We must also observe that this is different from the usual critical problems with a bi-Laplacian operator (−∆)2u = γu+ |u|p−1u, (1.7) EJDE-2021/52 BREZIS-NIRENBERG TYPE PROBLEMS WITH INVERSE OPERATOR 3 analyzed by Pucci-Serrin [12], under homogeneous Dirichlet boundary conditions, where the Sobolev’s critical exponent is pS = 2N N−4 ; see also the interesting book by Gazzola-Grunau-Sweers [8]. Note that (1.4) is not a variational problem. Therefore, to arrive at the desired results of existence of positive solutions applying any variational method we might easily apply the inverse operator (−∆)−1 of the Laplacian to problem (1.4), trans- forming it into the variational problem (1.1). Hence, establishing a direct connec- tion between the fourth-order equation (1.4) under homogeneous Navier boundary conditions and the non-local elliptic Dirichlet problem (1.1). Since the second order problem with the inverse operator (1.1) is of variational type, it has an associated energy functional Fγ(u) = 1 2 ∫ Ω |∇u|2dx− γ 2 ∫ Ω u(−∆)−1u dx− 1 p+ 1 ∫ Ω |u|p+1dx, (1.8) so that solutions of (1.1) can be obtained as critical points of the Fréchet-differentiable functional Fγ defined by (1.8). Here, as customary (−∆)−1u = v, if −∆v = u in Ω, v = 0 on ∂Ω. Thus, (−∆)−1 is a positive linear integral compact operator from L2(Ω) into itself, which is well-defined thanks to the Spectral Theorem. Moreover, we can rewrite the functional (1.8) as Fγ(u) = 1 2 ∫ Ω |∇u|2dx− γ 2 ∫ Ω |(−∆)−1/2u|2dx− 1 p+ 1 ∫ Ω |u|p+1dx. As a main result in this work, we prove the existence of positive solutions of problem (1.1) depending on the positive parameter γ. To do so, we first show the interval of the parameter γ for which there is the possibility of having positive solutions. Next, applying the well-known Mountain Pass Theorem (MPT for short) [4], we show that for the range 1 < p ≤ 2∗−1 there actually exists a positive solution to problem (1.1) provided 0 < γ < λ∗1, where λ∗1 is the first eigenvalue of the operator (−∆)2 under homogeneous Navier boundary conditions, i.e., λ∗1 = λ2 1 with λ1 being the first eigenvalue for the Lapla- cian under homogeneous Dirichlet boundary conditions. In the subcritical case (1 < p < 2∗ − 1) one might apply the MPT to (1.1) directly since, as we will show, our problem possesses the Mountain Pass (MP) geometry and, thanks to the compact embedding (1.5), the Palais-Smale condition is satisfied for the func- tional Fγ (see details below in Section 2). On the other hand, at the critical ex- ponent p = 2∗ − 1, the compactness of the Sobolev embedding is lost. Then, to check whether the Palais-Smale condition is satisfied becomes a delicate issue solving it. To overcome this lack of compactness we apply techniques based on the Concentration-Compactness Principle due to P.-L. Lions, [9], which allow us to prove the required Palais-Smale condition for N ≥ 7. Now we state the main results of this paper. Theorem 1.1. For every γ ∈ (0, λ∗1) there exists a positive solution u of problem (1.1) if: (i) 1 < p < 2∗ − 1 and N ≥ 3, (ii) p = 2∗ − 1 and N ≥ 7. 4 P. ÁLVAREZ-CAUDEVILLA, E. COLORADO, A. ORTEGA EJDE-2021/52 Surprisingly, even though our problem (1.1) is non-local but also a linear per- turbation of the problem (1.2), Theorem 1.1-(ii) addresses dimensions N ≥ 7, in contrast to the existence result of Brezis and Nirenberg about the linear pertur- bation (1.3), that covers the wider range N ≥ 4. In other words, the non-local term γ(−∆)−1u, despite of being just a linear perturbation, has an important ef- fect on the dimensions for which the classical Brezis-Nirenberg technique based on the minimizers of the Sobolev constant still works. Considering problem (1.7), Pucci-Serrin [12] introduced the concept of critical dimensions as the dimensions for which there are no positive solutions of (1.7) under homogeneous Dirichlet boundary conditions with γ > 0 arbitrarily small. They also conjectured that the critical dimensions for the corresponding problem with a polyharmonic operator (−∆)m are N = 2m + 1, . . . , 4m − 1. Combining the result of Pucci and Serrin [12] with existence theorems of Edmunds, Fortunato, Jannelli [6] and Noussair, Swanson, Yang Jianfu [10], the conjecture is true for m = 2. Moreover, because the connection between problem (1.1) and the fourth order problem (1.4), through Theorem 1.1-(ii) combined with Remark 2.5, we also prove the existence of positive solutions for (1.4). Note that in our case N = 7 is not a critical dimension anymore, in contrast with what happens for problem (1.7). It seems that the term (−∆)|u|p−1u behaves better than the pure power |u|p−1u with respect to the critical dimensions, despite having a worst behavior in the second order problem with the nonlocal term, (1.1). Alternative approach. We have also a relevant connection between problem (1.4) and a second order elliptic system through problem (1.1) to be explored at the end of the paper. In particular, taking w := (−∆)−1u, problem (1.1) provides us with the system −∆u = γw + |u|p−1u in Ω, −∆w = u in Ω, (u,w) = (0, 0) on ∂Ω, (1.9) which gives a different perspective to the problem in hand but providing similar results to the ones previously obtained in Theorem 1.1. However, since system (1.9) is not of variational type, as γ > 0, we can take v := √ γw in (1.9) and obtain the variational system −∆u = √ γv + |u|p−1u in Ω, , −∆v = √ γu in Ω, (u, v) = (0, 0) on ∂Ω, (1.10) whose associated energy functional is Jγ(u, v) = 1 2 ∫ Ω |∇u|2dx+ 1 2 ∫ Ω |∇v|2dx−√γ ∫ Ω uv dx− 1 p+ 1 ∫ Ω |u|p+1dx. (1.11) Remark 1.2. Thanks to the Maximum Principle, for a given u(x) as a positive solution of (1.1), and setting v = √ γ(−∆)−1u, it follows that v > 0. Thus, the pair (u, v) = (u, √ γ(−∆)−1u) is a positive solution of (1.10). And vice versa, given (u, v) a positive solution of (1.10) it is trivial that u(x) is a positive solution of (1.1). Although the equivalence between system (1.10) and the non-local problem (1.1) provides us with existence results for system (1.10) by means of Theorem 1.1, we prove independently the following. EJDE-2021/52 BREZIS-NIRENBERG TYPE PROBLEMS WITH INVERSE OPERATOR 5 Theorem 1.3. For every γ ∈ (0, λ∗1) there exists a positive solution (u, v) of system (1.10) if: (i) 1 < p < 2∗ − 1 and N ≥ 3, (ii) p = 2∗ − 1 and N ≥ 7. In the last section of the paper we extend our study to a higher order problem and prove, under analogous hypotheses, that there exists a positive solution of problem −∆u = γ(−∆)−mu+ |u|p−1u in Ω ⊂ RN , u = 0 on ∂Ω. (1.12) We recall the following well-known facts about polyharmonic operators of order 2m (m ≥ 1 an integer number) in a smooth domain Ω. The Navier boundary conditions for the operator (−∆)m are defined as u = ∆u = ∆2u = . . . = ∆k−1u = 0, on ∂Ω. Clearly, the operator (−∆)m is the m-th power of the classical Dirichlet Laplacian in the sense of the spectral theory and it can be defined as the operator whose action on a function u is given by 〈(−∆)mu, u〉 = ∑ j≥1 λmj |〈u1, ϕj〉|2, where (ϕi, λi) are the eigenfunctions and eigenvalues of the Laplace operator (−∆) with homogeneous Dirichlet boundary data. Thus, the operator (−∆)m is well defined in the space of functions that vanish on the boundary, Hm 0 (Ω) = { u = ∞∑ j=1 ajϕj ∈ L2(Ω) : ‖u‖Hm0 (Ω) = ( ∞∑ j=1 a2 jλ m j )1/2 <∞ } . Because of the lack of a comparison principle for higher-order equations, to obtain the existence results dealing with (1.12) we can not tackle this problem directly, and we need to use a similar correspondence to the one performed above for the problem (1.4), now with an elliptic system of m+ 1 equations. This article is organized as follows: In Section Section 2, we prove results for problem (1.1); and using similar ideas, for system (1.10) in Section 3. To finish, in Section 4 we study system (1.12). 2. Existence of positive solutions for (1.4) via problem (1.1) In this section we prove Theorem 1.1. First, we establish a condition on the range of values of γ necessary for the existence of positive solutions of (1.1). Let us consider the generalized eigenvalue problem associated with (1.1), −∆u = λ(−∆)−1u in Ω ⊂ RN , u = 0 on ∂Ω. (2.1) Then, we find that for the first eigenfunction ϕ1 associated with the first eigenvalue λ∗1 in (2.1),∫ Ω |∇ϕ1|2dx = λ∗1 ∫ Ω |(−∆)−1/2ϕ1|2dx, with ϕ1 ∈ H1 0 (Ω), 6 P. ÁLVAREZ-CAUDEVILLA, E. COLORADO, A. ORTEGA EJDE-2021/52 and, hence, λ∗1 = inf u∈H1 0 (Ω) ∫ Ω |∇u|2dx∫ Ω |(−∆)−1/2u|2dx . (2.2) On the other hand, it is clear that substituting the first eigenfunction of the Laplace operator under homogeneous Dirichlet boundary conditions, ϕ1, into (2.1), it fol- lows that λ∗1 = λ2 1. Thus, by definition of the powers of the Laplace operator, λ∗1 coincides with the first eigenvalue of the operator (−∆)2 under homogeneous Navier boundary conditions. Moreover, the first eigenfunction of (2.1) coincides with the first eigenfunction of the Laplace operator under homogeneous Dirichlet boundary conditions. Now, we prove the following result. Lemma 2.1. Problem (1.1) does not possess a positive solution when γ ≥ λ∗1. Proof. Assume that u is a positive solution to (1.1) and let ϕ1 be the principal eigenfunction of the Laplacian operator in Ω under homogeneous Dirichlet boundary conditions. Taking ϕ1 as a test function for the equation of (1.1) we obtain∫ Ω ϕ1(−∆)u dx = γ ∫ Ω ϕ1(−∆)−1u dx+ ∫ Ω |u|p−1uϕ1 dx > γ ∫ Ω ϕ1(−∆)−1u dx. Then, integrating by parts on both sides, λ1 ∫ Ω uϕ1 dx > γ ∫ Ω u(−∆)−1ϕ1 dx = γ λ1 ∫ Ω uϕ1 dx. Hence, γ < λ2 1 = λ∗1. � Lemma 2.2. The functional Fγ defined by (1.8) has the mountain pass geometry. Proof. Without loss of generality we can take a function g ∈ H1 0 (Ω) such that ‖g‖Lp+1(Ω) = 1. Then, taking a real number t > 0 and applying the Sobolev inequality (1.6) together with (2.2), we find that Fγ(tg) = t2 2 ∫ Ω |∇g|2dx− t2γ 2 ∫ Ω |(−∆)− 1 2 g|2dx− tp+1 p+ 1 ≥ t2 2 ( 1− γ λ∗1 )∫ Ω |∇g|2dx− tp+1 p+ 1 ≥ (1 2 ( 1− γ λ∗1 ) t2 − C (p+ 1) tp+1 )∫ Ω |∇g|2dx > 0, for t small enough, i.e., 0 < tp−1 < p+1 2C ( 1 − γ λ∗ 1 ) . Thus, the functional Fγ has a local minimum at u = 0, i.e. Fγ(tg) > Fγ(0) = 0, for any g ∈ H1 0 (Ω) provided t > 0 is small enough. Also, it is clear that Fγ(tg) = t2 2 ∫ Ω |∇g|2dx− γt2 2 ∫ Ω |(−∆)−1/2g|2dx− tp+1 p+ 1 ≤ t2 2 ‖g‖2H1 0 (Ω) − tp+1 p+ 1 . Then, Fγ(tg) → −∞ as t → ∞ and, thus, there exists û ∈ H1 0 (Ω) such that Fγ(û) < 0. � Now we turn our attention to the so-called Palais-Smale condition. EJDE-2021/52 BREZIS-NIRENBERG TYPE PROBLEMS WITH INVERSE OPERATOR 7 Definition 2.3. Let V be a Banach space. We say that a sequence {un} ⊂ V is a PS sequence for a functional F if F(un) is bounded and F′(un)→ 0 in V ′ as n→∞, (2.3) where V ′ is the dual space of V . Moreover, we say that a PS sequence {un} ⊂ V satisfies a PS condition if {un} has a convergent subsequence. (2.4) In particular, given a PS sequence {un} ⊂ V such that F(un) → c, if (2.4) is satisfied, we will say that the PS sequence satisfies the PS condition at level c for the functional F. Moreover, we say that the functional F satisfies the PS condition at level c if every PS sequence at level c for F possesses a convergent subsequence in V . For our problem, in the subcritical range the PS condition is always satisfied at any level c because of the compact Sobolev embedding (1.5). However, at the critical exponent 2∗ the problem is further complicated because of the lack of compactness in the Sobolev embedding. We will overcome this issue applying an argument based on the Concentration-Compactness Principle developed by P.-L. Lions, [9], proving that the functional Fγ satisfies the PS condition for levels c below a certain critical value c∗ (to be determined). Lemma 2.4. Let {un} be a PS sequence at level c for the functional Fγ , i.e. Fγ(un)→ c and F ′γ(un)→ 0 as n→∞. Then {un} is bounded in H1 0 (Ω). Proof. Since F ′γ(un)→ 0 in ( H1 0 (Ω) )′ , in particular 〈 F ′γ(un)| un ‖un‖H1 0(Ω) 〉 → 0. Thus, for any ε > 0 there exists a subsequence, denoted again by {un}, such that∫ Ω |∇un|2dx− γ ∫ Ω |(−∆)−1/2un|2dx− ∫ Ω |un|p+1dx = ‖un‖H1 0 (Ω) · o(1). Moreover, since Fγ(un)→ c, 1 2 ∫ Ω |∇un|2dx− γ 2 ∫ Ω |(−∆)−1/2un|2dx− 1 p+ 1 ∫ Ω |un|p+1dx = c+ o(1), for n large enough. Therefore, for a positive constant µ (to be determined below) we find that Fγ(un)− µ 〈 F ′γ(un)| un ‖un‖H1 0 (Ω) 〉 = c+ ‖un‖H1 0 (Ω) · o(1). That is(1 2 − µ ) ∫ Ω |∇un|2dx− γ (1 2 − µ ) ∫ Ω |(−∆)−1/2un|2dx− ( 1 p+ 1 − µ ) ∫ Ω |un|p+1dx = c+ ‖un‖H1 0 (Ω) · o(1). Hence, taking µ such that 1 p+1 < µ < 1 2 ,(1 2 − µ ) ∫ Ω |∇un|2dx− (1 2 − µ ) γ ∫ Ω |(−∆)−1/2un|2dx ≤ c+ ‖un‖H1 0 (Ω) · o(1), and using (2.2),(1 2 − µ )( 1− γ λ∗1 ) ∫ Ω |∇un|2dx ≤ (1 2 − µ ) ∫ Ω |∇un|2dx 8 P. ÁLVAREZ-CAUDEVILLA, E. COLORADO, A. ORTEGA EJDE-2021/52 + 2mu− (1 2 − µ ) γ ∫ Ω |(−∆)−1/2un|2dx ≤ c+ ‖un‖H1 0 (Ω) · o(1). From here, we conclude that(1 2 − µ )( 1− γ λ∗1 ) ‖un‖2H1 0 (Ω) ≤ c+ ‖un‖H1 0 (Ω) · o(1). Since 0 < γ < λ∗1, it follows that ( 1 2 − µ )( 1 − γ λ∗ 1 ) > 0 and, thus, because of the former inequality we conclude that the sequence {un} is bounded in H1 0 (Ω). � Proof of Theorem 1.1-(i). Let us consider the subcritical case 1 < p < 2∗−1. Given a PS sequence {un} ⊂ H1 0 (Ω) at level c, by Lemma 2.4 and the Rellich-Kondrachov Theorem the PS condition is satisfied. Hence, the functional Fγ satisfies the PS condition. Moreover, by Lemma 2.2 the functional Fγ possesses the MP geometry. Therefore, the hypotheses of the mountain pass theorem are fulfilled and we con- clude that the functional Fγ possesses a critical point u ∈ H1 0 (Ω). Moreover, if we define the set of paths Γ := {g ∈ C([0, 1], H1 0 (Ω)) : g(0) = 0, g(1) = û}, with û given as in the proof of Lemma 2.2, then Fγ(u) = c := inf g∈Γ max θ∈[0,1] Fγ(g(θ)). To show that u > 0, let us consider the functional, F+ γ (u) = Fγ(u+), where u+ = max{u, 0}. Repeating the arguments carried out above, with minor changes, one readily shows that what was proved for the functional Fγ still holds for the functional F+ γ . Therefore, u ≥ 0 and thanks to the Maximum Principle, u > 0. � Remark 2.5. Assuming that ∂Ω is a C2 manifold, by standard elliptic regularity theory, [7, Sec. 8.3, Theorem 1], it follows that u ∈ H1 0 (Ω) ∩H2(Ω) and thus, u is a positive weak solution of (1.4). 2.1. Concentration-compactness for the non-local problem (1.1). In this subsection we focus on the critical exponent case, p = 2∗ − 1, and our aim is to prove the PS condition for the functional Fγ . We carry out this task by means of a concentration-compactness argument based on the following lemma. Lemma 2.6 (P.-L. Lions,[9]). Let {un} be a weakly convergent sequence to u in H1 0 (Ω). Let µ, and ν be two nonnegative measures such that |∇un|2 → µ and |un|2 ∗ → ν as n→∞. Then, there exist a countable set I of points {xj}j∈I ⊂ Ω and some positive numbers µj, and νj such that |∇un|2 ⇀ µ = |∇u0|2 + ∑ j∈I µjδxj , |un|2 ∗ ⇀ ν = |u0|2 ∗ + ∑ j∈I νjδxj , (2.5) EJDE-2021/52 BREZIS-NIRENBERG TYPE PROBLEMS WITH INVERSE OPERATOR 9 where δxj is the Dirac’s delta centered at xj and satisfying µj ≥ SNν2/2∗ j . (2.6) Lemma 2.7. Assume p = 2∗ − 1. Then, the functional Fγ satisfies the Palais- Smale condition for any level c such that, c < c∗ = 1 N S N/2 N . Proof. Although the proof is rather standard we include it for the sake of com- pleteness. Let {un} ⊂ H1 0 (Ω) be a PS sequence of level c < c∗ for the functional Fγ . Thanks to Lemma 2.4, the sequence {un} is uniformly bounded and, as a consequence, we can assume that, up to a subsequence, un ⇀ u0 weakly in H1 0 (Ω), un → u0 strongly in Lq(Ω), 1 ≤ q < 2∗, un → u0 a.e. in Ω. (2.7) Next, for j ∈ I and ε > 0, let ϕj,ε ∈ C∞0 (Ω) be a cut-off function such that ϕj,ε = 1 in Bε(xj), ϕj,ε = 0 in Bc2ε(xj), |∇ϕj,ε| ≤ 2 ε , (2.8) where Br(xj) is the ball of radius r > 0, centered at a point xj ∈ Ω. Thus, using ϕj,εun as a test function we find that 〈F ′γ(un)|ϕj,εun〉 = ∫ Ω ∇un · ∇(ϕj,εun)dx− γ ∫ Ω ϕj,εun(−∆)−1un dx− ∫ Ω ϕj,ε|un|2 ∗ dx = ∫ Ω ϕj,ε|∇un|2dx− ∫ Ω ϕj,ε|un|2 ∗ dx+ ∫ Ω un∇un · ∇ϕj,εdx − γ ∫ Ω ϕj,εun(−∆)−1un dx. Moreover, from (2.5) and (2.7), we hve lim n→∞ 〈F ′γ(un)|ϕj,εun〉 = ∫ Ω ϕj,εdµ− ∫ Ω ϕj,εdν − γ ∫ Ω ϕj,εu0(−∆)−1u0dx + ∫ Ω u0∇u0 · ∇ϕj,εdx. By construction, lim ε→0 [ − γ ∫ Ω ϕj,εu0(−∆)−1u0dx+ ∫ Ω u0∇u0 · ∇ϕj,εdx ] = 0. Then, as F ′γ(un)→ 0 in ( H1 0 (Ω) )′ , we obtain that lim ε→0 (∫ Ω ϕj,εdµ− ∫ Ω ϕj,εdν ) = µj − νj = 0, and we conclude that νj = µj . (2.9) Finally, we have two options either the PS sequence has a convergent subsequence or it concentrates around some of the points xj . In other words, νj = µj = 0, or 10 P. ÁLVAREZ-CAUDEVILLA, E. COLORADO, A. ORTEGA EJDE-2021/52 there exists some νj > 0 such that, by (2.6) and (2.9), νj ≥ SN/2N . In case of having concentration, we find that c = lim n→∞ Fγ(un) = lim n→∞ Fγ(un)− 1 2 〈F ′γ(un)|un〉 = (1 2 − 1 2∗ ) ∫ Ω |u0|2 ∗ dx+ (1 2 − 1 2∗ ) νj ≥ 1 N S N/2 N = c∗, in contradiction with the hypotheses c < c∗. Therefore, the PS sequence has a convergent subsequence and the PS condition is satisfied. � It remains to show that we can obtain a path for Fγ under the critical level c∗. To obtain such a path we take test functions of the form ũε = Mφε, where φε = ϕj,R uj,ε, (2.10) with ϕj,R a cut-off function defined as (2.8) for some R > 0 small enough, M > 0 a large enough constant such that Fγ(ũε) < 0 and uj,ε are the family of functions uj,ε(x) = ( ε ε2 + |x− xj |2 )N−2 2 , (2.11) for ε > 0. Let us notice that the functions uj,ε are the extremal functions for the Sobolev’s inequality in RN , where the constant SN is achieved (see [13]). Then∫ RN |∇uj,ε|2dx = SN (∫ RN |uj,ε|p+1dx )2/2∗ . For the sake of simplicity we will consider xj = 0, we will denote ϕj,R = ϕ under the construction (2.8) and uj,ε = uε. We will also assume the normalization ‖uε‖L2∗ (Ω) = 1, (2.12) so that the Sobolev constant is given by SN = ∫ RN |∇uε|2dx. Then, under the previous considerations we define the set of paths Γε := {g ∈ C([0, 1], H1 0 (Ω)) : g(0) = 0, g(1) = ũε}, and consider the minimax values cε = inf g∈Γε max t∈[0,1] Fγ(g(t)). The final issue we must solve now is the fact that the levels cε are always below c∗ for ε small enough. To do so, we recall the following. Lemma 2.8 ([5, Lemma 1.1]). Let φ be the function denoted by (2.10) around the point xj = 0. Then ∫ RN φ2 εdx =  Cε+O(ε2) if N = 3, Cε2 2 | log ε|+O(ε2) if N = 4, Cε2 +O(εN−2) if N ≥ 5. (2.13) EJDE-2021/52 BREZIS-NIRENBERG TYPE PROBLEMS WITH INVERSE OPERATOR 11 Moreover, ‖∇φε‖22 = SN +O(εN−2). (2.14) Remark 2.9. Using similar arguments one could also estimate ‖φε‖L2∗ (Ω) ∼ C, however it is simpler if we normalize it as done in (2.12). To carry out the analysis of the levels cε we need some estimates dealing with the term ∫ Ω φε(−∆)−1φεdx. To this end, we prove the following result. Lemma 2.10. Let φε be the function denoted by (2.10) around the point xj = 0. Then, there exists a constant C > 0 independent of ε such that∫ Ω φε(−∆)−1φεdx > Cε1+ N N−4 and N ≥ 7. Proof. Let vε(x) = (−∆)−1φε(x) and note that because of the definition of the cut-off function (2.8), we can choose vε(x) such that (−∆)vε = φε in B2R(0), vε = 0 in ∂B2R(0). Moreover, since φε > 0 in B2R(0), thanks to the Maximum Principle, it follows that vε > 0 in B2R(0). Now, let us notice that for any x ∈ BR(0) we have φε(x) = uε(x) as well as ε− N−2 2( 1 + (Rε )2 )N−2 2 ≤ uε(x) ≤ ε− N−2 2 . Next, take ρ < R/2 and consider the function ṽ(x) = 2 N ( 1 − ( |x|2ρ )2 ) + , where (·)+ stands for the positive part. Then, ṽ satisfies the problem (−∆)ṽ = 1 ρ2 in B2ρ(0), ṽ = 0 in ∂B2ρ(0). To apply a comparison principle we choose ρ = ρ(ε) > 0, with ρ → 0 as ε → 0, such that (−∆)ṽ ≤ (−∆)vε in B2ρ(0). (2.15) Then, given ε > 0 arbitrarily small, we distinguish two cases depending upon ρ < ε or ρ > ε. In the first case, since uε(x) ∣∣ x∈B2ρ(0) ≥ ε− N−2 2( 1 + ( 2ρ ε )2 )N−2 2 > (1 5 )N−2 2 ε− N−2 2 . As a consequence if 1 ρ2 ≤ (1 5 )N−2 2 ε− N−2 2 , then (2.15) holds. Since ρ < ε, we have 5 N−2 2 ε N−2 2 ≤ ρ2 < ε2, (2.16) from which we conclude that N > 6. In the second case, ρ > ε, since uε(x) ∣∣ x∈B2ρ(0) ≥ ε− N−2 2( 1 + ( 2ρ ε )2 )N−2 2 = ε N−2 2 ρN−2 ( 4 + ( ερ )2 )N−2 2 > (1 5 )N−2 2 ε N−2 2 ρ2−N . 12 P. ÁLVAREZ-CAUDEVILLA, E. COLORADO, A. ORTEGA EJDE-2021/52 Once again, if 1 ρ2 ≤ (1 5 )N−2 2 ε N−2 2 ρ2−N . (2.17) Then (2.15) holds. By ρ > ε and (2.17), we have 1 ε4−N < 1 ρ4−N ≤ (1 5 )N−2 2 ε N−2 2 , (2.18) from which we deduce N > 6. Finally, by construction, 0 = ṽ(x) ∣∣ x∈∂B2ρ(0) < vε(x) ∣∣ x∈∂B2ρ(0) . Because of the Maximum Principle, we conclude that vε(x) > ṽ(x) for x ∈ B2ρ(0), thus ∫ Ω φε(−∆)−1φεdx ≥ ∫ BR(0) uε(x)vε(x)dx > ∫ B2ρ(0) uε(x)ṽ(x)dx ≥ ∫ Bρ(0) uε(x)ṽ(x)dx = 2 N ∫ Bρ(0) uε(x) ( 1− ( |x| 2ρ )2) dx ≥ 3 2N ∫ Bρ(0) uε(x)dx. On the other hand,∫ Bρ(0) uε(x)dx = ε− N−2 2 ∫ Bρ(0) 1( 1 + ( |x|ε )2 )N−2 2 dx = ε− N−2 2 ∫ ρ 0 rN−1( 1 + ( rε )2 )N−2 2 dr = ε N 2 ∫ ρ 0 ( r/ε )N−1( 1 + ( rε )2 )N−2 2 dr = ε N 2 +1 ∫ ρ/ε 0 sN−1( 1 + s2 )N−2 2 ds ≥ cεN2 +1 ∫ ρ/ε 0 sN−1ds = cε N 2 +1 (ρ ε )N , for a positive constant c. Then, we arrive at the estimate∫ Ω φε(−∆)−1φεdx > Cε− N−2 2 ρN and N > 6. (2.19) Now, from (2.16) and (2.18), we have the bounds ρ < ε or ρ ≤ (1 5 ) N−2 2(N−4) ε N−2 2(N−4) . (2.20) Since N > 6 it follows that N−2 2(N−4) < 1 so that, thanks to (2.20) and (2.19), we conclude ∫ Ω φε(−∆)−1φεdx > Cε1+ N N−4 and N > 6. � EJDE-2021/52 BREZIS-NIRENBERG TYPE PROBLEMS WITH INVERSE OPERATOR 13 Remark 2.11. Looking at the proof of Lemma 2.10 and considering either ρ < ε or ρ > ε we are forced to work with dimensions N ≥ 7. On the other hand, we might consider ρ = c0ε for a suitable constant in order to analyze if the case N = 6 could be included in the previous result. Then, following the previous argument in Lemma 2.10, such that ρ = c0ε and N = 6, we have uε(x) ∣∣ x∈B2ρ(0) ≥ ε−2( 1 + ( 2ρ ε )2 )2 = ( 1 1 + 4c20 )2 ε−2. As a consequence (2.15) holds if, 1 c20ε 2 ≤ ( 1 1 + 4c20 )2 ε−2, or equivalently if (1+4c20)2 ≤ c20, which is not possible. Therefore, as a consequence of that discussion, dimension N = 6 will be analyzed in a forthcoming paper by the use of different methods. Next we perform the analysis of the levels cε, proving that, in fact, the levels cε are always below the critical level c∗ provided ε > 0 is small enough. Lemma 2.12. Assume p = 2∗ − 1 and N ≥ 7. Then, there exists ε > 0 small enough such that sup 0≤t≤1 Fγ(tũε) < 1 N S N/2 N . Proof. Using (2.14) in Lemma 2.8 and assuming the normalization (2.12), we find that g(t) := Fγ(tũε) = t2M2 2 ‖∇φε‖2L2(Ω) − t2M2γ 2 ∫ Ω φε(−∆)−1φεdx− t2 ∗ M2∗ 2∗ = M2 2 ( SN +O(εN−2)− γF (ε) ) t2 − M2∗ 2∗ t2 ∗ , where F (ε) = ∫ Ω φε(−∆)−1φεdx. It is clear that limt→∞ g(t) = −∞ as well as that g(t) > 0 for t > 0 small enough. Therefore, the function g(t) possesses a maximum value at the point, tε := (M2 ( SN +O(εN−2)− γF (ε) ) M2∗ ) 1 2∗−2 . Moreover, at the point tε we have g(tε) = 1 N ( SN +O(εN−2)− γF (ε) )N/2 . Then, the proof will be complete if the inequality 1 N ( SN +O(εN−2)− γF (ε) )N/2 < 1 N S N/2 N , or, equivalently, the inequality O(εN−2) < γF (ε), (2.21) holds provided ε is small enough. Thanks to Lemma 2.10, we have F (ε) > Cε1+ N N−4 , so that (2.21) is equivalent to O(εN−2) < Cε1+ N N−4 , (2.22) 14 P. ÁLVAREZ-CAUDEVILLA, E. COLORADO, A. ORTEGA EJDE-2021/52 for ε > 0 small enough. So that it is sufficient to observe that (2.22) requires N − 2 > 1 + N N−4 which is equivalent to (N − 2)(N − 6) > 0, and that is obviously satisfied. � Proof of Theorem 1.1-(ii). Thanks to Lemmas 2.2 and 2.12, we find that 0 < cε ≤ sup 0≤t≤1 Fγ(tũε) < 1 N S N/2 N , provided ε > 0 is small enough. Because of Lemma 2.2 the functional Fγ has the mountain pass geometry. Moreover, because of Lemma 2.7 the functional Fγ satisfies the PS condition for any level cε provided ε > 0 is small enough. Therefore, we can apply the mountain pass theorem to obtain the existence of a critical point u ∈ H1 0 (Ω). The rest follows as in the subcritical case. � 3. Existence of positive solutions for (1.10) In this section we provide the existence result for the system (1.10). We start by stating the analogous results of those obtained for the functional Fγ . Lemma 3.1. The functional Jγ denoted by (1.11) has the mountain pass geometry. Proof. Let us consider, without loss of generality, a pair (g, h) ∈ H1 0 (Ω) × H1 0 (Ω) such that ‖g‖Lp+1(Ω) = 1. Then, taking a real number t > 0 and using the Young’s inequality together with the Poincaré’s inequality and the Sobolev inequality (1.6), we find that Jγ(tg, th) = t2 2 ∫ Ω |∇g|2dx+ t2 2 ∫ Ω |∇h|2dx− t2√γ ∫ Ω gh dx− tp+1 p+ 1 ≥ t2 2 ( ‖g‖2H1 0 (Ω) + ‖h‖2H1 0 (Ω) − √ γ ∫ Ω g2dx−√γ ∫ Ω h2dx ) − tp+1 p+ 1 ≥ t2 2 ( 1− √ γ λ1 )( ‖g‖2H1 0 (Ω) + ‖h‖2H1 0 (Ω) ) − ‖g‖2H1 0 (Ω) C p+ 1 tp+1 ≥ (1 2 ( 1− √ γ λ1 ) t2 − C p+ 1 tp+1 )( ‖g‖2H1 0 (Ω) + ‖h‖2H1 0 (Ω) ) , (3.1) where λ1 is the first eigenvalue of the Laplace operator under Dirichlet boundary conditions. Since 0 < γ < λ∗1 = λ2 1 it follows that √ γ < λ1 and we obtain( 1− √ γ λ1 ) > 0. Therefore, taking t > 0 such that 0 < tp−1 < p+ 1 2C ( 1− √ γ λ1 ) , from (3.1) we conclude that Jγ(tg, th) > 0. Thus, the functional Jγ has a local minimum at (u, v) = (0, 0), i.e., Jγ(tg, th) > Jγ(0, 0) = 0, for any pair (g, h) ∈ H1 0 (Ω) × H1 0 (Ω) provided t > 0 is small enough. Also, it is clear that, because of the Poincaré’s inequality, Jγ(tg, th) = t2 2 ∫ Ω |∇g|2dx+ t2 2 ∫ Ω |∇h|2dx− t2√γ ∫ Ω gh dx− tp+1 p+ 1 ≤ t2 2 ( ‖g‖2H1 0 (Ω) + ‖h‖2H1 0 (Ω) + √ γ ∫ Ω g2dx+ √ γ ∫ Ω h2dx ) − tp+1 p+ 1 EJDE-2021/52 BREZIS-NIRENBERG TYPE PROBLEMS WITH INVERSE OPERATOR 15 ≤ t2 2 ( 1 + √ γ λ1 )( ‖g‖2H1 0 (Ω) + ‖h‖2H1 0 (Ω) ) − tp+1 p+ 1 . Then Jγ(tg, th) → −∞ as t → ∞, and thus, there exists a pair (û, v̂) such that Jγ(û, v̂) < 0. � Lemma 3.2. Let {(un, vn)} ⊂ H1 0 (Ω) × H1 0 (Ω) be a PS sequence at level c for the functional Jγ , i.e. Jγ(un, vn) → c and J ′γ(un, vn) → 0, as n → ∞. Then {(un, vn)} is bounded in H1 0 (Ω)×H1 0 (Ω). Proof. Since J ′γ(un, vn)→ 0 in ( H1 0 (Ω)×H1 0 (Ω) )′ , in particular〈 J ′γ(un, vn)| (un, vn) ‖un‖H1 0 (Ω) + ‖vn‖H1 0 (Ω) 〉 → 0. Thus, for any ε > 0, there exists a subsequence, denoted again by {(un, vn)}, such that ∫ Ω |∇un|2dx+ ∫ Ω |∇vn|2dx− 2 √ γ ∫ Ω unvn dx− ∫ Ω |un|p+1dx = [ ‖un‖H1 0 (Ω) + ‖vn‖H1 0 (Ω) ] · o(1). Moreover, since Jγ(un, vn)→ c, 1 2 ∫ Ω |∇un|2dx+ 1 2 ∫ Ω |∇vn|2dx− √ γ ∫ Ω unvn dx− 1 p+ 1 ∫ Ω |un|p+1dx = c+ o(1), for n > 0 big enough. Therefore, for a positive constant µ (to be determined below) we find that Jγ(un, vn)−µ 〈 J ′γ(un, vn)| 1 ‖un‖H1 0 (Ω) (un, vn) 〉 = c+ [‖un‖H1 0 (Ω) +‖vn‖H1 0 (Ω)] · o(1). That is, (1 2 − µ )[ ∫ Ω |∇un|2dx+ ∫ Ω |∇vn|2dx− 2 √ γ ∫ Ω unvn dx ] − ( 1 p+ 1 − µ ) ∫ Ω |un|p+1dx = c+ [‖un‖H1 0 (Ω) + ‖vn‖H1 0 (Ω)] · o(1). Hence, taking µ such that 1 p+1 < µ < 1 2 ,(1 2 − µ )[ ∫ Ω |∇un|2dx+ ∫ Ω |∇vn|2dx ] − (1− 2µ) √ γ ∫ Ω unvn dx ≤ c+ [‖un‖H1 0 (Ω) + ‖vn‖H1 0 (Ω)] · o(1), and using Young’s inequality,(1 2 − µ )[ ∫ Ω |∇un|2dx+ ∫ Ω |∇vn|2dx− √ γ ∫ Ω u2 ndx− √ γ ∫ Ω v2 ndx ] ≤ c+ [‖un‖H1 0 (Ω) + ‖vn‖H1 0 (Ω)] · o(1). Then, because of the Poincaré inequality, we conclude that(1 2 −µ )( 1− √ γ λ1 ) [‖un‖2H1 0 (Ω)+‖vn‖2H1 0 (Ω)] ≤ c+[‖un‖H1 0 (Ω)+‖vn‖H1 0 (Ω)]·o(1), (3.2) 16 P. ÁLVAREZ-CAUDEVILLA, E. COLORADO, A. ORTEGA EJDE-2021/52 where λ1 is the first eigenvalue of the Laplace operator under Dirichlet boundary conditions. Since 0 < γ < λ∗1 = λ2 1, it follows that(1 2 − µ )( 1− √ γ λ1 ) > 0, and thus, by (3.2), we conclude that the sequence {(un, vn)} is bounded in H1 0 (Ω)× H1 0 (Ω). � Proof of Theorem 1.3-(i). If 1 < p < 2∗ − 1, given a PS sequence {(un, vn)} ⊂ H1 0 (Ω)×H1 0 (Ω) at level c, by Lemma 3.1, the functional Jγ has the MP geometry. Moreover, due to Lemma 3.2 and the compact inclusion H1 0 (Ω)×H1 0 (Ω) ↪→ Lp+1(Ω)× Lp+1(Ω), for 1 ≤ p < 2∗ − 1, provided by Rellich-Kondrachov Theorem, the functional Jγ satisfies the PS con- dition at any level c. Therefore, the hypotheses of the Mountain Pass Theorem are fulfilled and we conclude that the functional Jγ possesses a critical point (u, v) ∈ H1 0 (Ω)×H1 0 (Ω). Moreover, if we define the set of the paths Γ := { g ∈ C ( [0, 1], H1 0 (Ω)×H1 0 (Ω) ) : g(0) = (0, 0), g(1) = (û, v̂) } , with (û, v̂) given as in the proof of Lemma 3.1, then Jγ(u, v) = c := inf g∈Γ max θ∈[0,1] Jγ(g(θ)). To show the positivity of the pair (u, v) we argue as in the proof of Theorem 1.1-(i). Let us consider the functional J + γ (u, v) = Jγ(u+, v+), where, as before, u+ = max{u, 0}. Repeating with minor changes the arguments carried out above for the functional Jγ we conclude that the functional J + γ has a critical point (ũ, ṽ) such that ũ ≥ 0 and ṽ ≥ 0. Moreover, by the Maximum Principle, it follows that ũ > 0 and ṽ > 0, then (ũ, ṽ) is a positive solution of (1.10). � To prove the PS condition when p = 2∗ − 1 we must apply once again a concentration-compactness argument. Lemma 3.3. Assume p = 2∗−1. Then, the functional Jγ satisfies the Palais-Smale condition for any level c such that c < c∗ = 1 N S N/2 N . Proof. Let {(un, vn)} ⊂ H1 0 (Ω) × H1 0 (Ω) be a PS sequence of level c < c∗ for the functional Jγ . Thanks to Lemma 3.2, the sequence {(un, vn)} is uniformly bounded and, as a consequence, we can assume that there exists a subsequence still denoted by {(un, vn)}, such that (un, vn) ⇀ (u0, v0) weakly in H1 0 (Ω)×H1 0 (Ω), (un, vn)→ (u0, v0) strongly in Lq(Ω)× Lq(Ω), 1 ≤ q < 2∗, (un, vn)→ (u0, v0) a.e. in Ω. (3.3) Moreover, we can assume that, up to a subsequence, there exist three measures µ, µ̃ and ν such that |∇un|2, |∇vn|2 and |un|2 ∗ , converge in the sense of the measures EJDE-2021/52 BREZIS-NIRENBERG TYPE PROBLEMS WITH INVERSE OPERATOR 17 µ, µ̃ and ν respectively. Thus, because of Lemma 2.6, there is a countable set I of points {xj}j∈I ⊂ Ω, and some positive numbers µj , µ̃j and νj such that |∇un|2 ⇀ dµ = |∇u0|2 + ∑ j∈I µjδxj , |∇vn|2 ⇀ dµ̃ = |∇v0|2 + ∑ j∈I µ̃jδxj , |un|2 ∗ ⇀ dν = |u0|2 ∗ + ∑ j∈I νjδxj , (3.4) where δxj is the Dirac’s delta centered at xj with j ∈ I and satisfying µj ≥ SNν2/2∗ j . (3.5) Next, for j ∈ I, let ϕj,ε ∈ C∞0 (Ω) be a cut-off function satisfying (2.8) centered at xj ∈ Ω. Thus, using (ϕj,εun, ϕj,εvn) as a test function, we find, 〈J ′γ(un, vn)|(ϕj,εun, ϕj,εvn)〉 = ∫ Ω ∇un · ∇(ϕj,εun)dx+ ∫ Ω ∇vn · ∇(ϕj,εvn)dx− 2 √ γ ∫ Ω ϕj,εunvn dx − ∫ Ω ϕj,εu 2∗ n dx = ∫ Ω ϕj,ε|∇un|2dx+ ∫ Ω ϕj,ε|∇vn|2dx− ∫ Ω ϕj,εu 2∗ n dx+ ∫ Ω un〈∇un,∇ϕj,ε〉dx + ∫ Ω vn〈∇vn,∇ϕj,ε〉dx− 2 √ γ ∫ Ω ϕj,εunvndx. Moreover, from (3.3) and (3.4), we have lim n→∞ 〈J ′γ(un, vn)|(ϕj,εun, ϕj,εvn)〉 = ∫ Ω ϕj,εdµ+ ∫ Ω ϕj,εdµ̃− ∫ Ω ϕj,εdν − 2 √ γ ∫ Ω ϕj,εu0v0dx + ∫ Ω u0〈∇u0,∇ϕj,ε〉dx+ ∫ Ω v0〈∇v0,∇ϕj,ε〉dx. By construction, lim ε→0 [ − 2 √ γ ∫ Ω ϕj,εu0v0dx+ ∫ Ω u0 〈∇u0,∇ϕj,ε〉 dx+ ∫ Ω v0 〈∇v0,∇ϕj,ε〉 dx ] = 0. Then, as J ′γ(un)→ 0 in ( H1 0 (Ω)×H1 0 (Ω) )′ , we obtain that lim ε→0 (∫ Ω ϕj,εdµ+ ∫ Ω ϕj,εdµ̃− ∫ Ω ϕj,εdν ) = µj + µ̃j − νj = 0, and we conclude that νj = µj + µ̃j . (3.6) Finally, we have two options either the PS sequence has a convergent subsequence or it concentrates around some of the points xj . In other words, νj = µj = µ̃j = 0, or there exists some νj > 0 such that, by (3.5) and (3.6), νj ≥ S N/2 N . In case of having concentration, we find that c = lim n→∞ Jγ(un, vn) 18 P. ÁLVAREZ-CAUDEVILLA, E. COLORADO, A. ORTEGA EJDE-2021/52 = lim n→∞ Jγ(un, vn)− 1 2 〈Jγ(un, vn)|(un, vn)〉 = (1 2 − 1 2∗ ) ∫ Ω |u0|2 ∗ dx+ (1 2 − 1 2∗ ) νj ≥ 1 N S N/2 N = c∗, in contradiction with the hypotheses c < c∗. Therefore, the PS sequence has a convergent subsequence and the PS condition is satisfied. � Next we show that we can obtain a path for Jγ under the critical level c∗. To obtain such a path we will assume test functions of the form (ũε, ṽε) = (Mφε,Mρφε), where φε = ϕj,R uj,ε, with ϕj,R is a cut-off function defined by (2.8), for some R > 0 small enough, M > 0 a sufficiently large constant such that Jγ(ũε, ṽε) < 0, ρ is a positive term to be determined below and uj,ε are the family of functions defined by (2.11). For the sake of simplicity, in the sequel we will consider xj = 0 as well as the normalization (2.12). Then, under the previous construction, we define the set of paths Γε := { g ∈ C ( [0, 1], H1 0 (Ω)×H1 0 (Ω) ) : g(0) = (0, 0), g(1) = (ũε, ṽε) } , and consider the minimax value cε = inf g∈Γε max t∈[0,1] Jγ(g(t)). Now we prove that, in fact, the levels cε are always below c∗ for ε > 0 small enough. Lemma 3.4. Assume p = 2∗ − 1. Then, there exists ε > 0 small enough such that sup 0≤t≤1 Jγ(tũε, tṽε) < 1 N S N/2 N , provided N ≥ 7. Proof. Let us denote by F (ε) the estimate (2.13) in Lemma 2.8. Then, assuming the normalization (2.12), g(t) :=Jγ(tũε, tṽε) = ( t2M2 2 + ρ2t2M2 2 ) ‖∇φε‖2L2(Ω) − t 2M2ρ √ γ ∫ Ω φ2 εdx− t2 ∗ M2∗ 2∗ = t2M2 2 ( (1 + ρ2)[SN +O(εN−2)]− 2ρ √ γF (ε) ) − t2 ∗ M2∗ 2∗ . It is clear that limt→∞ g(t) = −∞, therefore, the function g(t) possesses a maximum value at the point tε = (M2[(1 + ρ2)[SN +O(εN−2)]− 2ρ √ γF (ε)] M2∗ ) 1 2∗−2 . Moreover, at the point tε, g(tε) = 1 N [( 1 + ρ2 ) [SN +O(εN−2)]− 2ρ √ γF (ε) ]N/2 . Then, the proof will be complete if we can choose ρ > 0 such that the inequality, [(1 + ρ2)[SN +O(εN−2)]− 2ρ √ γF (ε)] < SN , (3.7) EJDE-2021/52 BREZIS-NIRENBERG TYPE PROBLEMS WITH INVERSE OPERATOR 19 holds provided ε > 0 is small enough. Indeed, if we take ρ = εα, with α > 0 (to be determined), inequality (3.7) is equivalent to SNε 2α +O(εN−2+2α) +O(εN−2) < 2 √ γεαF (ε), Since SNε 2α +O(εN−2+2α) +O(εN−2) = O(ετ ) with τ = min{2α,N − 2 + 2α,N − 2} = min{2α,N − 2}, we are left to prove α > 0 can be chosen such that O(ετ ) < 2 √ γεα ·  Cε+O(ε2), if N = 3, Cε2 2 | log ε|+O(ε2), if N = 4, Cε2 +O(εN−2), if N ≥ 5. (3.8) provided ε > 0 is small enough. If N = 3, the corresponding inequality in (3.8) holds true if τ = min{2α, 1} > α+ 1 that is not possible. If N = 4, the corresponding inequality (3.8) holds true if O(ετ ) < C √ γε2α+2| log ε| ⇒ O(ετ−2−α) < C √ γ| log ε|, and thus, necessarily τ = min{2α, 2} > 2 + α, that, once again, is not possible. If N ≥ 5, the corresponding inequality (3.8) holds true if τ = min{2α,N − 2} > 2 + α. Let us observe that min{a, b} = 1 2 (a+ b− |a− b|), hence, inequality (3.8) will be satisfied if we can choose α > 0 such that N − |2α− (N − 2)| > 6. (3.9) Now we have two options, either 2α > N − 2 or 2α < N − 2. • In the first case, thanks to inequality (3.9), we find the condition N 2 + 1 > N − α > 4, that can be fulfilled only for N > 6. • In the second case, thanks to inequality (3.9), we find the condition N−2 > 2α > 4, that can be fulfilled, once again, only for N > 6. Thus, if N ≥ 7 we can choose α > 2 such that (3.8) is satisfied. Finally, note that with the assumption ρ = εα we have tε = (M2[(1 + ρ2)[SN +O(εN−2)]− 2ρ √ γF (ε)] M2∗ ) 1 2∗−2 ≥ δ > 0, provided ε > 0 is small enough. � Proof of Theorem 1.3-(ii). Applying Lemmas 3.1 and 3.4, we find that 0 < cε ≤ sup 0≤t≤1 Jγ(tũε, tṽε) < 1 N S N/2 N , provided ε > 0 is small enough. Indeed, due to Lemma 3.1 the functional Jγ has the MP geometry. Moreover, thanks to Lemma 3.3 the functional Jγ satisfies the PS condition for any level cε with ε > 0 small enough. Therefore, we can apply the Mountain Pass Theorem and conclude the existence of a critical point (u, v) ∈ H1 0 (Ω)×H1 0 (Ω). The rest follows as in the subcritical case. � 4. Polyharmonic problems Let us consider the higher order problem with generalized Navier boundary con- ditions, (−∆)m+1u = γu+ (−∆)m|u|p−1u in Ω ⊂ RN (−∆)ju = 0 for 0 ≤ j ≤ m, on ∂Ω (4.1) 20 P. ÁLVAREZ-CAUDEVILLA, E. COLORADO, A. ORTEGA EJDE-2021/52 with m a natural number bigger than 1, and the variational problem obtained applying the operator (−∆)−m to (4.1), −∆u = γ(−∆)−mu+ |u|p−1u in Ω ⊂ RN u = 0 on ∂Ω (4.2) associated with the energy functional Fγ,m(u) = 1 2 ∫ Ω |∇u|2dx− γ 2 ∫ Ω |(−∆)−m/2u|2dx− 1 p+ 1 ∫ Ω |u|p+1dx. Note that, as it happens for m = 1, the embedding features for problem (4.2) are governed by the standard second-order equation, −∆u = |u|p−1u, thus, the variational framework coincides with the one of the case m = 1, so that we also consider 1 < p ≤ 2∗ − 1. Let us observe that if we try to prove the existence of a positive solution problem (4.2) directly, as performed for the problem (1.1) in Section (2), we immediately run into complications. Because of the lack of a comparison principle, we can not use a similar argument to Lemma (2.10) when dealing with the operator (−∆)−m. Thus, we will use the correspondence between problem (4.2) and the elliptic system −∆u = γ 1 m+1 v1 + |u|p−1u, −∆v1 = γ 1 m+1 v2, −∆v2 = γ 1 m+1 v3, . . . −∆vm = γ 1 m+1u  in Ω (u, v1, . . . , vm) = (0, 0, . . . , 0) on ∂Ω (4.3) whose associated energy functional is Jγ,m(U) = 1 2 ∫ Ω |∇u|2dx+ 1 2 m∑ i=1 ∫ Ω |∇vi|2dx− γ 1 m+1 m+ 1 (∫ Ω uv1 dx+ ∫ Ω uvm dx + m−1∑ i=1 ∫ Ω vivi+1dx ) − 1 p+ 1 ∫ Ω |u|p+1dx, (4.4) where U = (u, v1, . . . , vm). The functional Jγ,m has the same structure as the functional Jγ thus, the ideas developed in Section 3 will fit, with slight variations, in this scenario. Let us denote by Λ∗1 the first eigenvalue of the operator (−∆)m+1 under the homogeneous generalized Navier boundary conditions given in (4.1). It is clear from the spectral definition of the operator (−∆)m+1 that Λ∗1 = λm+1 1 with λ1 the first eigenvalue of the Laplace operator under homogeneous Dirichlet boundary conditions. The aim of this last section is then to prove the following result. Theorem 4.1. Assume 1 < p < 2∗ − 1. Then, for every γ ∈ (0,Λ∗1), there exists a positive solution to system (4.3). EJDE-2021/52 BREZIS-NIRENBERG TYPE PROBLEMS WITH INVERSE OPERATOR 21 Theorem 4.2. Assume p = 2∗ − 1. Then, for every γ ∈ (0,Λ∗1), there exists a positive solution to system (4.3) provided N ≥ 7. We start determining the interval of values of the parameter γ > 0 compatible with existence of positive solutions related to problem (4.2). Lemma 4.3. Equation (4.2) does not possess a positive solution when γ ≥ Λ∗1. Proof. Using as a test function in (4.2) the first eigenfunction ϕ1 associated with the first eigenvalue λ1 for the Laplacian operator (−∆) with homogeneous Dirichlet boundary conditions together with Λ∗1 = λm+1 1 the result follows. � Next we deal with the mountain pass conditions. We state the analogous results to those of the case m = 1. Since the proofs of the next results rely on the ideas developed for the case m = 1, we will only remark the main differences, if any. Lemma 4.4. The functional Jγ,m (U) has the mountain pass geometry. The proof of the above lemma is similar to that of Lemma 3.1 so we omit it. Lemma 4.5. Let Em := H1 0 (Ω)×H1 0 (Ω)× . . .×H1 0 (Ω) and {Un} = {(un, v1,n, . . . , vm,n)} ⊂ Em be a PS sequence for the functional Jγ,m, i.e., Jγ,m(Un)→ c and J ′γ,m(Un)→ 0 as n→∞. Then {Un} is bounded in Em. Proof. Arguing as in the proof of Lemma 3.2 we find that (m+ 1) (1 2 − µ )( 1− 2γ 1 m+1 (m+ 1)λ1 )( ‖un‖2H1 0 (Ω) + m∑ i=1 ‖vi,n‖2H1 0 (Ω) ) ≤ (m+ 1)c+ ( ‖un‖H1 0 (Ω) + m∑ i=1 ‖vi,n‖H1 0 (Ω) ) · o(1). Keeping in mind Lemma 4.3, it follows that(1 2 − µ )( 1− 2γ 1 m+1 (m+ 1)λ1 ) > 0, and we conclude the boundedness of the sequence {Un} in Em. � Proof of Theorem 4.1. Combining Lemmas 4.4, 4.5, the Rellich-Kondrachov Theo- rem, the hypotheses of the MPT are fulfilled and we arrive at the same conclusion as in the proof of Theorem 1.3-(i). � To finish, we deal with the critical case p = 2∗ − 1. As it was done in previous sections, with the aid of a concentration-compactness argument we will prove that the PS condition is satisfied for any level below the critical level c∗ = 1 N S N/2 N . Let us observe that the critical level c∗ is independent of the order of the inverse operator involved in problem (4.2) as it coincides with the critical level for problem (1.1). Lemma 4.6. The functional Jγ,m defined by (4.4) satisfies the Palais-Smale con- dition for any level c below the critical level c∗. 22 P. ÁLVAREZ-CAUDEVILLA, E. COLORADO, A. ORTEGA EJDE-2021/52 Proof. Let {Un} = {(un, v1,n, . . . , vm,n)} ⊂ Em be a PS sequence of level c < c∗. Because of Lemma 4.5 and Lemma 2.6, we can replicate the steps of the proof of Lemma 3.3 incorporating the slight difference that, instead (3.6), we find now νj = µj + m∑ i=1 µ̃i,j . (4.5) with µj ≥ SNν2/2∗ j . (4.6) Then, either the PS sequence has a convergent subsequence or it concentrates around some of the points xj . In other words, νj = µj = µ̃i,j = 0, or there exists some νj > 0 such that, thanks to (4.5) and (4.6), νj ≥ S N/2 N . In case of having concentration, c = lim n→∞ Jγ,m(Un) = lim n→∞ Jγ,m(Un)− 1 2 〈Jγ,m(Un)|Un〉 = (1 2 − 1 2∗ ) ∫ Ω |u0|2 ∗ dx+ (1 2 − 1 2∗ ) νj ≥ 1 N S N/2 N = c∗, in contradiction with the hypotheses c < c∗. � Finally, we show that we can obtain a path for the functional Jγ,m under the critical level c∗. Following the ideas of the previous sections, we will assume test functions of the form Ũε = (ũε, ṽ1,ε, . . . , ṽm,ε) = (Mφε,Mρφε, . . . ,Mρφε), (4.7) with M > 0 a sufficiently large constant so that Jγ,m(Ũε) < 0, ρ is positive term to be determined as in the case m = 1, and uj,ε are the family of functions defined by (2.11). As performed above we will consider xj = 0. Then, under the previous construction, let us define the set of paths Γε := {g ∈ C([0, 1],Em) : g(0) = 0, g(1) = Ũε}, and consider the minimax value cε = inf g∈Γε max t∈[0,1] Jγ,m(g(t)). Next, we check that any level cε is always below c∗ provided ε > 0 is small enough. This is done applying Lemma 2.8. Lemma 4.7. Assume p = 2∗−1 and N ≥ 7. Then, there exists ε > 0 small enough such that sup 0≤t≤1 Jγ,m(tŨε) < 1 N S N/2 N . Proof. Let us denote by F (ε) estimate (2.13) in Lemma 2.8. Then, assuming the normalization (2.12), we obtain g(t) :=Jγ,m(tŨε) = (1 2 (1 +mρ2)[SN +O(εN−2)]− γ 1 m+1 m+ 1 (2ρ+ (m− 1)ρ2)F (ε) ) M2t2 EJDE-2021/52 BREZIS-NIRENBERG TYPE PROBLEMS WITH INVERSE OPERATOR 23 − M2∗ t2 ∗ 2∗ . Proceeding as in the proof of Lemma 3.4, the proof will be completed if we can choose ρ > 0 such that the inequality O(εN−2) +mρ2SN +mρ2O(εN−2) < 2 γ 1 m+1 m+ 1 (2ρ+ (m− 1)ρ2)F (ε), holds provided ε > 0 is small enough. We take ρ = εα with α > 0 (to be determined) and τ = min{N − 2, 2α, 2α+N − 2} = min{N − 2, 2α}. Then, since O(εα + ε2α) = O(εα), we are left to prove that for a constant C > 0 the inequality O(ετ ) < CεαF (ε), (4.8) holds provided ε > 0 is small enough. Since inequality (4.8) coincides with (3.8) the arguments performed in Lemma 3.4 allow us to complete the proof. � Proof of Theorem 4.2. Thanks to Lemmas 3.1 and 3.4, we find that cε ≤ sup t≥0 Jγ(tŨε) < 1 N S N/2 N , provided ε > 0 is sufficiently small. Hence, combining Lemmas 4.4 and 4.6 we can apply the mountain pass theorem and conclude the existence of a critical point U ∈ Em. The rest follows as in the former cases. � Acknowledgments. This research was partially supported by the Ministry of Economy and Competitiveness of Spain, and by the FEDER under research projects MTM2016-80618-P and PID2019-106122GB-I00. P. Álvarez-Caudevilla was also supported by the Ministry of Economy and Competitiveness of Spain under re- search project RYC-2014-15284. References [1] P. Álvarez-Caudevilla, J. D. Evans, V. A. Galaktionov; Countable families of solutions of a limit stationary semilinear fourth-order Cahn–Hilliard-type equation I. Mountain pass and Lusternik-Schnirel’man patterns in RN . 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Pablo Álvarez-Caudevilla Departamento de Matemáticas, Universidad Carlos III de Madrid, Av. Universidad 30, 28911, Leganés, Madrid, Spain Email address: pacaudev@math.uc3m.es Eduardo Colorado Departamento de Matemáticas, Universidad Carlos III de Madrid, Av. Universidad 30, 28911, Leganés, Madrid, Spain Email address: ecolorad@math.uc3m.es Alejandro Ortega Departamento de Matemáticas, Universidad Carlos III de Madrid, Av. Universidad 30, 28911, Leganés, Madrid, Spain Email address: alortega@math.uc3m.es 1. Introduction 2. Existence of positive solutions for (??) via problem (??) 2.1. Concentration-compactness for the non-local problem (??) 3. Existence of positive solutions for (??) 4. Polyharmonic problems Acknowledgments References