Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains

The paper is concerned with the existence and multiplicity of positive solutions of the nonhomogeneous Choquard equation over an annular type bounded domain. Precisely, we consider the following equation \[ -\De u = \left(\int_{\Om}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u+f \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a smooth bounded annular domain in $\mathbb{R}^N( N\geq 3)$, $2^*_{\mu}=\frac{2N-\mu}{N-2}$, $f \in L^{\infty}(\Om)$ and $f \geq 0$. We prove the existence of four positive solutions of the above problem using the Lusternik-Schnirelmann theory and varitaional methods, when the inner hole of the annulus is sufficiently small.


Introduction
In the pioneering work, Tarantello [31] studied the nonhomogeneous elliptic equation −∆u = |u| * − u + f in Ω, u = on ∂Ω, (1.1) where * = N N− is the critical Sobolev exponent and Ω is a bounded domain in R N with smooth boundary. If f ∈ H − then it is shown that there exists at least two solutions of (1.1) by using variational methods. Cao and Zhou [9] proved the existence of two positive solutions of the following nonhomogeneous elliptic equation where f (x, u) is a Carathéodory function with subcritical grotwh at ∞. Further, many researchers investigated (1.1) and (1.2) for the existence and multiplicity of solutions. For details, we refer [10,11,20,21,33] and references therein. Recently, Gao and Yang [30] proved the existence of two positive solutions of the nonhomogeneous Choquard equation involving Hardy-Littlewood-Sobolev critical exponent using the splitting Nehari manifold method of Tarantello [31].
The existence, uniqueness, and multiplicity of positive solutions of the nonlocal elliptic equation, precisely the Choquard equation both for mathematical analysis and in perspective of physical models has recently gained signi cant attention amongst researchers. As an instance, in 1954 Pekar [28] proposed the equation to study the quantum theory of polaron. Later in 1976, Ph. Choquard [22] examined the steady state of one component plasma approximation in Hartee-Fock theory using (1.3). In [22], Leib proved the existence and uniqueness of the ground state of (1.3). The work of Moroz and Schaftingen enriches the literature of Choquard equations. In [25] authors studied the following Choquard equation where α ∈ ( , N), N ≥ , Iα is the Riesz Potential and F(u) ∈ C (R, R) with sub critical growth. In this work authors established the existence of ground state soloutions of (1.4) and assuming some suitable growth conditions on F and V, they studied the properties like constant sign solutions and radial symmetry of the solution. Moreover, authors proved the Pohožaev identity and nonlocal Brezis-Kato type estimate. Interested readers are referred to [16,24,26,27] and references therein for the study of Choquard equation on the unbounded domain.
Concerning the boundary value problems of Choquard equation, Gao and Yang [15] studied the Brezis-Nirenberg type existence results for the following critical equation where λ > , < µ < N, h(u) = u, Ω is a smooth bounded domain in R N . Later in [14] authors proved the existence and multiplicity of positive solutions for convex and convex-concave type nonlinearities (h(u) = u q , < q < ) using variational methods.
The geometry of the domain Ω plays an essential and signi cant role on the existence and multiplicity of the elliptic boundary value problems. Indeed, in [12], Coron proved the existence of a high energy positive solution of the problem −∆u = |u| * − u in Ω, u = on ∂Ω, (1.5) where Ω is a bounded domain in R N (N ≥ ), precisely an annulus with a small hole. Later in [3], Bahri and Coron, proved that a positive solution always exists as long as the domain has non-trivial homology with Z -coe cients. In [6], Benci and Cerami studied the following equation in Ω, u = on ∂Ω, (1.6) where ε ∈ R + , Ω is a bounded domain in R N (N ≥ ) and f : R+ → R is a C , function. Here authors proved that there exists ϵ * > such that for all ε ∈ ( , ϵ * ), (1.6) has cat(Ω)+ solutions under some growth conditions on the function f . Since then, the study of existence and multiplicity of solutions of elliptic equations over non-contractible domain has been substantially studied, for instance, [4,5,13,20,29,32] and references therein. The existence of high energy solution of (1.5) is a much more delicate issue. In this spirit, recently Goel, Rădulescu and Sreenadh [19] studied the Coron problem for Choquard equations. Here authors proved the existence of a positive high energy solution for the problem (P f ) when f (x) ≡ and Ω is a smooth bounded domain in R N (N ≥ ) satisfying the following condition (A) There exists constants < R < R < ∞ such that In the light of above works, in this article, we study following problem where * µ = N−µ N− , is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality (2.1) and f ∈F withF := {f : f ∈ L ∞ (Ω), f ≥ , f ≢ }. The domain Ω ⊂ R N (N ≥ ) satis es the condition (A). Here we prove the existence of four solutions of the problem (P f ). To achieve this, we rst seek the help of Nehari manifold associated with (P f ) to prove the existence of the rst solution (say u ). To proceed further, we prove many new estimates on the convolution terms involving the minimizers of best constant S H,L (see Lemma 4.1,4.3 and 4.4 ). With the help of these estimates we prove that the minima of the functional over N f is below the rst critical level where the rst critical level is Here J f is the energy functional associated to (P f ) (de ned in (2.3)). Moreover, J f satis es the Palais-Smale condition below the rst critical level. Subsequently, we show the existence of the second and the third solution of (P f ), in N − f (a closed subset of the Nehari manifold) by using a well-known result of Ambrosetti [2](see Lemma 5.2) and assumption (A). To study the existence of the fourth solution, a high energy solution, we prove that the functional J f satis es the Palais-Smale condition between the rst and the second critical levels, where the second critical level is To prove the existence of fourth solution, we use the minmax Lemma (See Lemma 6.6). To the best of our knowledge, there is no work on the existence and multiplicity of solutions to Choquard equations (P f ) in non-contractible domains. With this introduction, we state our main result. Theorem 1.1. Assume µ < min{ , N}, f ∈ L ∞ (Ω) and f ≥ and Ω be a bounded domain satisfying the conditon (A). Then there exists e * > such that (P f ) has at least three positive solutions whenever < f H − < e * . Moreover, if R is small enough then there exists e ** > such that (P f ) has at least four positive solutions whenever < f H − < e ** .
The paper is organized as follows: In Section 2, we give the variational framework and preliminary results. In section 3, using the Nehari manifold technique, we prove the existence of the rst solution. In section 4, we prove some crucial estimates of the minimizer of S H,L (de ned in (2.2)) and analyze the Palais-Smale sequences. In section 5, we prove the existence of the second and third solution. In section 6, we prove the existence of the fourth solution.

Variational framework and preliminary results
We start with the familiar Hardy-Littlewood-Sobolev Inequality which leads to the study of nonlocal Choquard equation using variational methods. Proposition 2.1. [23](Hardy-Littlewood-Sobolev Inequality) Let t, r > and < µ < N with /t+µ/N+ /r = , f ∈ L t (R N ) and h ∈ L r (R N ). There exists a sharp constant C(t, r, µ, N) independent of f and h such that

Equality holds in (2.1) if and only if f ≡ (constant)h and
for some A ∈ C, ≠ γ ∈ R and a ∈ R N .
The best constant for the embedding Consequently, we de ne where C > is a xed constant , a ∈ R N and b ∈ ( , ∞) are parameters. Moreover, Lemma 2.3. [15] For N ≥ and < µ < N. Then The energy functional J f : H (Ω) → R associated with the problem (P f ) is where u + = max(u, ). By using Hardy-Littlewood-Sobolev inequality (2.1), we have It is not di cult to show that the functional J f ∈ C (H (Ω), R) and moreover, if µ < min{ , N} then J f ∈ C (H (Ω), R). Since J f is not bounded below on H (Ω), it is worth to consider the Nehari manifold where , denotes the usual duality. We de ne Note that when f (x) ≡ , Υ (Ω) is independednt of Ω and Υ (Ω) : Notations: Throughout the paper we will use the notation J = J, N = N, dxdy.
An easy consequence of (2.1) gives J f is coercive and bounded below on N f . Proposition 2.6. For any u, v ∈ H (Ω), we have Proof. For details of the proof, see [ Remark 2.8. We remark that by [15,Lemma 1.3], S H,L is never achieved on bounded domain. Therefore if u is a solution of the following equation Claim: Tn H − = (DΥ ) + on( ). Let ψ ∈ H (Ω) such that ψ = then by Lemma 2.7, we know that there exists a t > such that (2.5) Taking into account (2.4), (2.5), Proposition 2.6 and employing Hölder's inequality, for each n, we have For any ψ ∈ H (Ω) with ψ = , we have Clearly, N f contains every non zero solution of (P f ) and we know that the Nehari manifold is closely related to the behavior of the bering maps ϕu : R + → R de ned as ϕu(t) = J f (tu). It is easy to see that tu ∈ N f if and only if ϕ u (t) = and elements of N f correspond to stationary points of the bering maps. It is natural to divide N f into the following sets We also denote the in mum over N + f and N − f as

Existence of First Solution
In this section we prove the existence of rst solution by showing the existence of minimizer for J f over the Nehari manifold N f . First we state some Lemmas whose proof can be found in [30]. We further prove some properties of the manifold N + f .
sequence for J f , then there exists a subsequence of {un}, still denoted by {un}, and a non-zero u ∈ H (Ω) such that un → u strongly in H (Ω). Moreover, u ∈ N f and is a solution to (P f ).
Proof. J f is bounded below and coercive implies {un} is bounded in H (Ω). So, there exists a subsequence still denoted by {un} such that un u weakly in H (Ω). By [19,Lemma 4.2], we have J f (u ) = . In particular, fu dx. Now, using the fact that a is weakly lower semi continuous we have − Ω fwn dx = on( ) and since Ω fwn dx = on( ), we get a(wn) = on( ). Hence un → u strongly in H (Ω).
Lemma 3.8. If u be a solution of (P f ) then u ∈ C (Ω). Moreover, u is a positive solution.
Proof. Let u be a solution of (P f ) and Then by the standard elliptic regularity u ∈ C (Ω). Since f ≥ , we get u ≥ and by using strong maximum principle, u is a positive solution of (P f ). Proof.

Asymptotic estimates and Palais-Smale Analysis
In this section we shall prove that the functional J f satis es Palais-Smale condition strictly below the rst critical level and (strictly) between the rst and second critical levels. To start with, we shall prove several new estimates on the nonlinearity.
It is known from Lemma 2.2 that the best constant S H,L is achieved by the function which is a solution of the problem −∆u = (|x| −µ * |u| We may assume R = ρ, R = /ρ for ρ ∈ ( , ). Now, de ne υρ ∈ C ∞ c (R N ) such that ≤ υρ(x) ≤ for all x ∈ R N , radially symmetric and
(ii) Result follows from the de nition of J and by (i).
It yields a contradiction. Hence results follows. Proof.

(i) Consider
(4.4) From the de nition of u σ ϵ , we have the following estimates Therefore, from above estimates and (4.4), we obtain desired result.
(ii) Consider where J i are de ned in (4.3). Using the Hardy-Littlewood-Sobolev inequality and the de nition of ξϵ, we have the following estimates: Now using the same estimates as above we can easily obtain → as ρ → and completes the proof.

N−µ+ + O(ϵ N ) and now using this and Hardy-Littlewood-Sobolev in-
This proves part (ii). Now to prove part (iii), consider Now we will give a Lemma which is taken from [18]. For the sake of completeness, we provide a complete proof.
where u is the local minimum obtained in Lemma 3.10.
Proof. We will divide the proof in two cases: Case 1: * µ > . It is easy to see that there exists A > such that (a + b) p ≥ a p + b p + pa p− b + Aab p− for all a, b ≥ and p > , Case 2: < * µ ≤ . We recall the inequality from [7,Lemma 4]: there exist C(depending on * µ ) such that, for all a, b ≥ , Consider |x − y| µ dxdy, for all u ∈ H (Ω) and i = , , , .
Employing (4.5), we have the following inequality: where A ϵ is sum of eight non-negative integrals and each integral has an upper bound of the Then utilizing the de nition of O , u ∈ L ∞ (Ω) and Hardy-Littlewood-Sobolev inequality, we have By the choice of s we have In a similar manner, we have Once again using (4.5), we have the following inequality: where A ϵ is sum of eight non-negative integrals and each integral has an upper bound of the form By the similar estimates as in Subcase 1, de nition of O , the fact that tg ϵ,σ ρ ∈ H (Ω) and regularity of u , we have Write (u (y)) * µ − = (u (y)) r .(u (y)) s with * µ − = r + s, < + s < * µ . Then utilizing the de nition of O , u ∈ L ∞ (Ω) and Hardy-Littlewood-Sobolev inequality, we have By the choice of s we have Subcase 3: when (x, y) ∈ O . Using (4.5), we have where A ϵ is sum of eight non-negative integrals and each integral has an upper bound of the form where A ϵ is sum of eight non-negative integrals and each integral has an upper bound of the form By the similar estimates as in Subcase 2, we have where u is the local minimum in Lemma 3.10.
Proof. By Lemma 3.8, u ∈ L ∞ (Ω) and u > in Ω. This implies Claim 1: There exists a R > such that Clearly, For any ϵ < − ρ there exists c > such that − ϵ > c > ρ so we get ).
This proves the claim 1. Now using Lemma 4.4, we have for all Θ < . Taking Θ = * µ , we have This on utilizing Lemma 4.3 and claim 1 gives . Moreover there exists a t > such that for su ciently small ϵ > we have tϵ > t . Clearly the function is an increasing function in [ , S H,L (ϵ)]. Therefore, Hence there exits a ϵ > such that for every < ϵ < ϵ we have Lemma 4.6. The following holds: (iii) For each < ϵ ≤ ϵ , there exists t > and such that u + t g ϵ,σ ρ ∈ U .
(iii) First, we will show that there exists a constant c > such that < t − u +tg ϵ,σ ρ u +tg ϵ,σ Hence, J f (t − (un)un) → −∞ as n → ∞, contradicts the fact that J f is bounded below on N f . Therefore, It implies that u + t g ϵ,σ ρ ∈ U .
At this point we will state Global compactness Lemma for the functional J f which is a version of Theorem 4.4 of [19].
(iii) Using part (ii), we obtain the following estimate for each u ∈ Σ and < ω < Using (5.1) in part (i) we get Therefore, we get Thus, there exists e > such that Υ − f (Ω) > whenever f H − < e where v is a minimizer of S H,L , λ n ∈ R + , y n ∈ Ω. Moreover, if n → ∞ then λ n → , y n |y n | → y is the unit vector in R N . Thus we obtain where the function u + s t g ϵ,σ ρ de ned in Lemma 4.6.
Note that from Lemma 5.5, G is well de ned.
Equivalently, (P f ) have another two di erent solutions which are di erent from u .
Proof. Using Lemma 5.8 and Lemma 5.3, we have Now the proof follows from Lemma 4.8(i) and Lemma 5.2.

Existence of Fourth solution
In this section we will prove the existence of high energy solution by using Brouwer's degree theory and minmax theorem given by Brezis and Nirenberg [8].
It implies that < η ≤ |β(h ϵn ρ ) − σ| ≤ Cϵn → + as ϵn → + , a contradiction. Proof of Theorem 1.1 : First note that by Lemma 3.8, we have all solutions of (P f ) are positive in Ω and from Lemma 3.7, we have u ∈ N + f ⊂ H (Ω) such that J f (u ) = Υ f whenever < f H − < e . By Proposition 5.9 we have two more critical point u , u ∈ N − f of J f such that in J f (u ), J f (u ) < Υ f (Ω) + N−µ+ ( N−µ) S N−µ N−µ+ H,L . Therefore we get three positive solutions of (P f ) whenever < f H − < e * where e * is de ned in Proposition 5.9. Let e ** = min{e * , e * } then by Proposition 6.12, we get u ∈ N − f J f (u ) = γ f .