MULTIPLE SOLUTIONS FOR A CRITICAL QUASILINEAR EQUATION WITH HARDY POTENTIAL

. In this paper, we investigate the following quasilinear equation involving a Hardy potential:

Dj(aij(u)Diu)+ 1 2 N i,j=1 a ij (u)DiuDju− µ |x| 2 u= au + |u| 2 * −2 u in Ω, u= 0 on ∂Ω, where 2 * = 2N N −2 is the Sobolev critical exponent for the embedding of H 1 0 (Ω) into L p (Ω), a > 0 is a constant and Ω ⊂ R N is an open bounded domain which contains the origin. We will prove that under some suitable assumptions on a ij , when N ≥ 7 and µ ∈ [0, µ * ) for some constant µ * , problem (P) admits an unbounded sequence of solutions. To achieve this goal, we perform the subcritical approximation and the regularization perturbation.
where Ω ⊂ R N is an open bounded domain which contains the origin, 2 * = 2N N −2 is the Sobolev critical exponent for the embedding of H 1 0 (Ω) into L p (Ω), a > 0 is a constant and µ ∈ [0, µ * ) for some constant µ * , which will be determined later. Note that µ|x| −2 u is the Hardy potential.
When a ij (t) = δ ij and µ = 0, problem (1) is reduced to the classical Brezis-Nirenberg problem: 1978 FENGSHUANG GAO AND YUXIA GUO admits infinitely many solutions. The work of [15] inspired a lot of efforts in the last decade to generalize similar existence and multiplicity results to other classes of equations. For instance, Cao, Peng and Yan [7] proved the existence of infinitely many solutions for p-Laplacian equation, while Guo, Liu and Wang [17] investigated a quasilinear equation. Further more, Cao and Yan [8] considered (1) with a ij (t) = δ ij but µ = 0. They proved that if a ij (t) = δ ij , N ≥ 7, µ ∈ [0, (N −2) 2 4 − 4), problem (1) has infinitely many solutions.
From the physical point of view, equations with Hardy potential arise from many physical contexts, such as molecular physics [20], quantum cosmology [3] and linearization of combustion models [16]. On the other hand, from the mathematical point of view, the main reason of interest in Hardy potentials lies in their criticality. More precisely, we see that although the Hardy term has the same homogeneity as the Laplacian operator, it does not belong to the Kato's class and hence can not be regarded as a lower order perturbation term.
In the present paper, we will show the existence of infinitely many solutions for a more general critical quasilinear equation with Hardy potential. Here the critical exponent reads as 2N N −2 . In general, for example, when a ij (t) = (1 + |t| 2α )δ ij , α ≥ 0, the corresponding critical exponent is (1 + α) 2N N −2 . Note that if R N u 2α |∇u| 2 < +∞ for α > 0, then the Hardy integral term R N u 2 |x| 2 is compact. In this case, the Hardy term has no effect to the equation. Therefore, we are only concerned with the case when α = 0, that is the case when a ij is bounded. For more general cases, such as the case when a ij (t) = (1 + |t| 2α )δ ij with α > 0, we could obtain the similar existence results.
(3) However the energy functional is continuous but is not of C 1 in H 1 0 (Ω), even for the subcritical problems (i.e., the critical exponent 2 * is replaced by a smaller exponent q < 2 * ). Hence the standard critical point theory can not be applied directly. To overcome this difficulty, we apply a regularization approach proposed by Liu and Wang (e.g., [21,25,26,27]. On the other hand, since 2 * is the critical exponent for the Sobolev embedding from H 1 0 (Ω) to L p (Ω), the second difficulty we are facing is that the energy functional I(u) does not satisfy the Palais.Smale condition (P.S. condition in short) for large energy levels. The third difficulty is that, unlike in [17], every nontrivial solution of equation (1) is singular at x = 0 if µ = 0 (see [5]). So, different techniques are needed to deal with the case µ = 0.
We assume that a ij (s) satisfies the following conditions: ), a ij = a ji , and there exists a constant C > 0 such that |a ij | ≤ C, |D s a ij | ≤ C for all (x, s) ∈Ω × R.
(A 2 ) there exist α, β > 0 such that for all ξ ∈ R N , s ∈ R, it holds |s|→∞ a ij (s) exist, which will be denoted as A ij .
To overcome the difficulty caused by the critical exponent, we follow the idea of [15] and proceed an approach of subcritical approximation. More precisely, we consider the following perturbed problem: where ε → 0.
, µ * = min{μ − 1, α(μ − 4)}, the main results of the present paper are the following: Let {u n } be a sequence of solutions to the subcritical problems (4) with ε = ε n → 0, and there exists a constant C independent of n such that u n ≤ C. Then for any a > 0, 0 < µ < µ * , {u n } converges strongly in H 1 0 (Ω) as n → ∞.
As a consequence of Theorem 1.1, we have the following multiplicity result.
Before the end of this introduction, let us outline the methods of the proofs. By Theorem 1.1 in [14], (4) admits a sequence of solutions {u ε,l }. Then we will study as ε → 0, the convergence of solutions {u ε,l } to the solutions u l of (1). For this purpose, one of our main ingredients is to show that the weak solutions of the subcritical problem (4) satisfy a variational inequality, and this allows us to adapt some of the arguments from [6] and [8]. For the convergence analysis, we first show a concentration result of solutions to the subcritical problems, then by using a decomposition result introduced by C. Tintarev and K. Fineseler [31,32], we establish some integral estimates, which results in a finer control for the solutions near the blow up points. Moreover, as we mentioned before, due to the appearance of the Hardy term, the solution u ε,l has singularity at the origin and is not in L ∞ anymore, so new ideas are needed to establish the local Pohozaev identity in small balls. Finally an application of the local Pohozaev identity rules out the possibility of bubble blow-ups which in turn gives the strong convergence result.
The paper is organized as follows. In Section 2, we discuss the concentration compactness of the solutions to the subcritical problems, while in section 3 and 4, according to the blow up locations given in Section 2, we investigate the bounds of the approximation solutions in the safe regions. In Section 5, we establish a local Pohozaev identity which together with the estimates in Sections 4 allows us to give the proofs of Theorem 1.1 and 1.2. Some essential integral estimates for a linear problem in divergent form are put in the Appendices.
Throughout this paper, without additional declarations, we will denote α, β as in (A 2 ), the norms of H 1 0 (Ω) and L p (Ω) by · and · p respectively.

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FENGSHUANG GAO AND YUXIA GUO 2. Concentration compactness analysis. In this section, we will consider the concentration behaviors for solutions of problem (4) as ε → 0. The following result is important for us.
The next lemma concerns the convergence of {u n } to the solution of (1).
Lemma 2.2. Suppose {u n } is a sequence of solutions of the problem (4) with ε = ε n → 0, as n → ∞. If u n u in H 1 0 (Ω) as n → ∞, then u is a solution of (1). Proof. This result can be proved by following the same arguments as in Lemma 2.2 in [17] (where µ = 0).
By (A 2 ) and (A 5 ), we have Lemma 2.3. Assume {u n } is a sequence of solutions of (4) with ε = ε n → 0 and u n u in H 1 0 (Ω). Then Proof. For any T > 0, denote u T = u, if |u| ≤ T and u T = ±T , if ±u ≥ T . We first prove that u T n → u T in H 1 0 (Ω) as n → ∞. In fact, taking u T and u T n as test functions in (1) and (4) respectively, we have Since u n → u in L 2 * −1 , we have Ω |u n | 2 * −2−εn u n + au n + µu n |x| 2 u T n → Ω |u| 2 * −2 u + au + µu |x| 2 u T , as n → ∞.
Lemma 2.4. Assume {u n } is a sequence of solutions of (4) with ε = ε n and u n u in Proof. Without loss of generality, we may assume v n w in . And Lemma 2.4 can be proved by using the similar arguments as in the proof of Lemma 2.3.
To proceed, let us recall some known facts from [31,32]. For a function u ∈ D 1,2 (R N ), we may define the dilation and translation of u by g λ,y u = λ N −2 2 u(λ(· − y)) for y ∈ R N , λ ∈ R + . Consider the transformation group of dilations and translations: By Theorem 2.2 and 3.2 in [31], we have the following profile decomposition results: such that Now we assume ε n → 0 and u n is a weak solution of (4) with ε = ε n . We also suppose that u n u in H 1 0 (Ω). Then by Lemma 2.2, we have that u is a solution of (1). In the following, we regard u n , u as the elements of the space , it is also bounded in D 1,2 (R N ) and has the profile decomposition (9), and if g k n = g λ n,k ,x n,k , the assertion (2) in (9) is equivalent to Moreover, since {u n } is bounded in H 1 (R N ), we also have ( see [32]) λ n,k → ∞, as n → ∞.
Lemma 2.5. Suppose that u n has a profile decomposition (9). Then V = |U k | satisfies one of the following three differential inequalities: Proof. Set L = lim n→∞ λ n,k dist(x n,k , ∂Ω), then there are two cases.
where A is a constant. Note that (g k . By Lemma 2.4, for the left hand side of (14), we have On the other hand, we may estimate the right hand side of (14) as follows: where x 0 denotes the limitation of {λ n,k x n,k } n when it is finite, otherwise the above term goes to 0.
Combining the above arguments, we deduce that Note that for any δ > 0, there exists an A δ > 0 such that for any s > 0, it holds that we obtain the desired result (11) and (12) for V = |U k |.
Case 2. L < ∞. Without loss of generality, we assume x n,k → x * ∈ ∂Ω as n → ∞ and the inner normal at x * is the Ox N axis. We take ϕ ∈ C ∞ 0 (R N L ). Then for n large enough, we still have ψ = g k n ϕ = λ . Therefore, we can use the similar arguments as in the Case 1 to complete the proof of (13).
Take Kelvin transformṽ(y) = 1 Using Morser iteration, we can show thatṽ(y) , thus the result holds for V . And if (13) holds, we can obtain the desired results by using a similar argument.
Proof. Indeed, under the assumptions that V = |U k | satisfies one of (11), (12) and (13), we can always deduce that either

FENGSHUANG GAO AND YUXIA GUO
In the first situation, by assumption (A 2 ) and Hardy inequality, we have Thus, there exists a constant C > 0 such that We obtain the desired result from (3) in (9). And in the second case one can prove it similarly.
In the following, we will establish some integral estimates for the approximation solution u n under the norm · * ,p1,p2,λ . As we mentioned before, due to the appearance of the Hardy potential, the solutions are not in L ∞ (Ω), fortunately, by using Morse iteration, we could have some L p (for some p ) estimates, which turned out to be enough for our proof. Proposition 1. Suppose u n is a solution of (4) with ε = ε n → 0, and has a decomposition in form of (9). Denote λ n = inf k {λ n,k }, for any p 1 , there is a constant C, depending on p 1 , p 2 , such that u n * ,p1,p2,λn ≤ C.
In order to prove Proposition 1, we need a few lemmas. Indeed with the aid of Lemma 2.1, we only need to prove these lemmas in a simple form.
αμ−µ , with p 2 < 2 * < p 1 , let q i given by Then there is a constant C = C(p 1 , p 2 ) such that for any λ > 0, and then the result follows.
Proof of Proposition 1. By Lemma 3.3, the result is true for some p 1 , p 2 close to 2 * with p 2 < 2 * < p 1 . Using the bootstrap Lemma 3.2, we can prove Proposition 1 for some pairs of We complete the proof.
4. The estimates in the safe domains. Suppose {u n } have decomposition forms of (9), and assume λ n = λ n,1 = inf{λ n,k }, x n = x n,1 . By Corollary 1, the number of the bubbles of u n is finite, we can find a constantc > 0 independent of n, such that the region does not contain any concentration points of u n , for any n. We call this region a safe region for u n . Let Proposition 2. Suppose {u n } is a sequence of solutions for the subcritical problem (4) with ε = ε n , and u n u in H 1 0 (Ω) as ε n → 0. Then there exist constants r 0 and C r0 with 0 < r 0 < 1 and C r0 = C(r 0 ) > 0 such that where x ∈ A 2 n , 0 < r ≤ r 0 , and p 1 , p are constants satisfying n , does not contain any concentration points of u n , we can deduce that By Lemma 2.1, we see that v n = |u n | satisfies the following differential inequality For ∀x ∈ A 2 n , 0 < r < R ≤ 1, let y = λ n z ∈ Ω}. We claim that if p is a constant satisfying By the estimates above, we may assume By iteration, we have for 2 ≤ p < Notice that where κ = p−2 p−1 . So, ṽ n L p (Dr(y)) ≤ 1 2 ṽ n L p (D R (y)) + C ṽ n L 1 (D R (y)) , ∀1 ≥ R > r > 0.
For the case (26), then by using the similar arguments, we can prove the desired results.

FENGSHUANG GAO AND YUXIA GUO
By Hölder inequality, we have Combining (27) and (28), we have for We complete the proof.
Proof. Applying Proposition 2, there exists an r 0 such that for r ≤ r 0 n . We choose ϕ = η 2 v n as a test function in (5), then Ω |Du n | 2 η 2 ≤ C Ω u 2 n |Dη| 2 + C Ω (|u n | 2 * −εn−1 + au n )u n η 2 . Since which implies that We have three cases to consider: (i) B n Ω c = ∅; (ii) B n ⊂ Ω and 0 / ∈B n ; (iii) B n ⊂ Ω and 0 ∈B n . In each case, the point x * is chosen as follows: In case (i), we take x * ∈ Ω c , with |x * − x n | ≤ (c + 8)λ − 1 2 n and n(x − x * ) ≤ 0 on ∂ e D n . With this x * , we can check that x · x * ≥ 0 in B n . In case (ii), we take a point x * = x n , then x · x * ≥ 0 in B n . In case (iii), we take a point x * = 0. Thus, in each case, we have x · x * ≥ 0 in B n . Lemma 5.1. Let p n = 2 * − ε n , then u n satisfies the following local Pohozaev identity: Proof. Let f (s) = |s| pn−2 s + as, F (s) = 1 pn |s| pn + 1 2 as 2 . Multiplying the equation Taking ϕ = u n η as the test function in (4), we have Notice that η = 0 on ∂D n ∂B n , u n = 0 and D k u n = ∂un ∂n n k on ∂ e D n = ∂D n ∂Ω, We have a ij (un)DiunDjununη Dn |un| pn η+a Dn |un| 2 η+ In case (i) and (ii), u n ∈ C 2 (D n ), thus (29) is the usual local Pohozaev identity. Now we prove that (29) holds in case (iii). Let D n,θ = D n \ B θ (0), then we have a ij (un)DiunDjununη aij(un)DiunDjunx k n k ηdσ.
By the choice of η, the integral are indeed over A 2 n , thus a Dn u 2 n η ≤ C A 2 n |Du n | 2 + |Du n ||u n |λ 1 2 n + |u n | pn + |u n | 2 , from Proposition 2 and Corollary 2, for p 1 > 2 * , we have that On the other hand, let D n = D λ −1 n (x n ), recall the decomposition of u n , u n = u 0 + u n,1 + u u,2 , with u n,2 → 0, as n → ∞. For n large, We have Moreover, for µ <μ−1, we have 2( √μ + √μ − µ) > N , so U j ∈ L 2 (R N ), if µ <μ−1. By the definition of u n,1 , and since λ n is the slowest concentration rate of blow-up, there is a constant C > 0 such that Thus LHS ≥ Cλ −2 n . And where σ is a small constant. This can be achieved if µ α <μ − 4, then with this p 1 , 2 < N −2 2 − N p1 , we obtain a contradiction.
In order to prove Theorem 1.2, we first look at the existence of solutions to the subcritical problem (4), under assumptions (A 1 ) − (A 5 ). Following the idea of [14] (see also [17]), we apply the method of regularity approximation. For reader's convenience, we give the sketch of the proof.
Next, similar as in [17], we may choose two functionals J andJ independent of ε n and θ such that This is possible, for example, we may take where α, β are the constants in the condition (A 2 ), σ is any positive constant, c σ andC are some positive constants, and p ∈ (2, 2 * ) is a fixed constant. Without loss of generality, we may assume α − μ µ = 1. Let 0 < λ 1 ≤ λ 2 ≤ · · · be the eigenvalue of Laplacian with Dirichlet boundary condition on ∂Ω. Assume m is the least integer such that λ m+1 > a. Let E m be the direct sum of the eigenspaces corresponding to λ 1 , λ 2 , · · · , λ m . Set B k the unit ball in R m+k , we define the critical value by: where Then we deduce similarly as Lemma 6.2 in [17], there exists a constant σ 0 > 0 such that By the symmetry mountain theorem, we see that c n k (θ) is a critical value of I εn,θ . Suppose u n k (θ) is the corresponding critical point, by (34) and (39), I εn,θ (u n k (θ)) ≤ b k and u n k (θ) ≤ M k , for some constant M k . Let c n k = lim θ→0 c n k (θ), then σ 0 ≤ c n k ≤ b k . By the convergence theorem in [14], we have u n k (θ) → u n k in H 1 0 (Ω), I εn,θ (u n k (θ)) → I εn (u n k ) = c n k , as θ → 0. And u n k is a weak solution of (4) with ε = ε n with u n k ≤ M k . Proof of Theorem 1.2. It is sufficient to prove the existence of infinitely many solutions of u to the problem (1). Indeed, for any k and n, suppose that c n k is the critical value of the subcritical problem with ε = ε n . By (39), we see that By Theorem 1.1, we have u n k → u k in H 1 0 (Ω), I εn (u n k ) → I(u k ) = c k , as n → ∞. And u k is a weak solution of (1).
In order to prove c k → ∞, as k → ∞, we will compare c k with the corresponding minimax values of the functional J. Note that the functional J is associated to the semilinear problem. Let Since J ≤ I εn,θ ≤J, E k ⊂ G k , we have hence But d k → ∞ (see Lemma 6.3 in [17]). We finish the proof of Theorem 1.2.
Appendix A. L p estimates.
If u ∈ H 1 (R N ) satisfies that There exists a small δ > 0, such that if Bτ (x0) |g| Proof. The proof is similar to that of the Lemma A.4 in [8]. Thus we omit it.