Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent

In this paper, we are concerned with the following nonlinear Schrodinger equations with hardy potential and critical Sobolev exponent 
\begin{equation}\label{eq0.1} 
\left\{\begin{array}{ll} 
-\Delta u+\lambda a(x)u^q=\mu\frac{u}{|x|^{2}}+|u|^{2^{*}-2}u,& \textrm{in}\, \mathbb{R}^N, \\ 
 u>0, & \textrm{in}\,\mathcal{D}^{1,2}(\mathbb{R}^N), （1） 
\end{array} 
 \right. 
\end{equation} 
where $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent, $0\leq \mu<\overline{\mu}=\frac{(N-2)^2}{4}$, $a(x)\in C(\mathbb{R}^N)$. We first use an abstract perturbation method in critical point theory to obtain the existence of positive solutions of （1） for small value of $|\lambda|$. Secondly, we focus on an anisotropic elliptic equation of the form 
\begin{equation}\label{eq0.2} 
-{\rm div}(B_\lambda(x)\nabla u)+\lambda a(x)u^q=\mu\frac{u}{|x|^{2}}+|u|^{2^*-2}u, x\in\mathbb{R}^N. （2） 
\end{equation} 
The same abstract method is used to yield existence result of positive solutions of （2） for small value of $|\lambda|$.


(Communicated by Susanna Terracini)
Abstract. In this paper, we are concerned with the following nonlinear Schrödinger equations with hardy potential and critical Sobolev exponent where 2 * = 2N N −2 is the critical Sobolev exponent, 0 ≤ µ < µ = (N −2) 2 4 , a(x) ∈ C(R N ). We first use an abstract perturbation method in critical point theory to obtain the existence of positive solutions of (1) for small value of |λ|. Secondly, we focus on an anisotropic elliptic equation of the form − div(B λ (x)∇u) + λa(x)u q = µ u |x| 2 + |u| 2 * −2 u, x ∈ R N . ( The same abstract method is used to yield existence result of positive solutions of (2) for small value of |λ|.

JING ZHANG AND SHIWANG MA
We notice that by a variant of the Pohozaev identity (see [6]), for any solution u of problem (3), we have where (.|.) denotes the scalar product in R N . As a consequence, if a(x) ≡ 1, (4) gives R N u q+1 dx = 0, and thus problem (3) has no solution if λ = 0. There are several elliptic problems on R N which are perturbative in nature. For these perturbative problems a specific approach, that takes advantage of such a perturbative setting, seems the most appropriate. These abstract tools are provided by perturbation methods in critical point theory. Actually, it turns out that such a framework can be used to handle a large variety of equations, uaually considered different in nature.
The main reason of interest in Hardy term relies in their criticality, indeed they have the same homogeneity as the Laplacian and the critical Sobolev exponent and don't belong to the Kato class, hence they cannot be regarded as a lower order perturbation term.
Another reason why we investigate (3), in addition to the inverse square potential, is the presence of the critical Sobolev exponent and the unbounded domain R N , which cause the loss of compactness of embedding D 1,2 (R N ) → L 2 * (R N ) and H 1 (R N ) → L p (R N ). Hence, including the non-compactness of the imbedding we face a type of triple loss of compactness whose interacting each other will result in some new difficulties. In last two decades, loss of compactness leads to many interesting existence and nonexistence phenomena for elliptic equations.
Ambrosetti et al. [2] studied the equation by applying a perturbative approach, which relies on a suitable use of an abstract perturbation method in critical point theory discussed in [1,3,4]. Cingolani [9] used perturbative approach to study the positive solutions to perturbed elliptic problems in R N involving critical Sobolev exponent, then obtained that the equation has a positive solution.
In this paper, we shall prove the existence of positive solutions of (3), assuming that a(x) satisfies the following assumptions: As in [9], we shall apply the perturbative approach discussed in [1,3,4]. Roughly speaking, for small value of |λ|, problem (3) can be seen as perturbation of the critical problem on R N : Denote β = √ µ − µ, Catrina and Wang [8] proved that for 0 < µ < µ, ε > 0, all positive solutions of (6) are of the form z ε (x) = ε for an appropriate constant C N > 0. These solutions achieve S µ , where Firstly, we are motivated by a paper due to Silvia Cingolani [9] and obtain the following result: Then there exist λ > 0 and ε > 0 such that for any λ ∈ R, |λ| ≤ λ, problem (3) has a positive solution u λ and u λ → z ε in D 1,2 (R N ) as λ → 0.
Secondly, we focus on an anisotropic elliptic equation of the form where N ≥ 3 and B λ (x) = I + λB(x), B(x) is a symmetric matrix with positive bounded coefficients and a(x) satisfies (a 1 )−(a 2 ). The same abstract method is used to yield existence result of positive solutions of (8) for small value of |λ|. We point out that a perturbation result for an anisotropic Schrödinger equation is obtained in [5] in the subcritical case.
2. Critical points for perturbed functionals. In this section, we shall recall an abstract perturbative method in critical point theory, which provides an abstract tool to deal with several perturbed semilinear equations and to find multiple homoclinic orbits to a class of second-order Hamiltonian systems (for example, see [3,4,9]). We consider a family of C 2 functionals f λ defined on a Hilbert space E of the form f λ (u) = f 0 (u) + λg(u), where f 0 , g ∈ C 2 (E, R) and λ > 0. On the unperturbed functional f 0 we shall make the following assumption: (I) there exists a d-dimensional C 2 manifold Z, d ≥ 1, consisting of critical points of f 0 . Such Z is called critical manifold of f 0 . The presence of a critical manifold is usually due to the fact that the unperturbed functional f 0 is invariant under the action of a symmetry group: for example, in the case discussed in the next sections it will be invariant under scale changes.
Let T z Z denote the tangent space to Z. We further suppose: Assumption (III) is a sort of nondegeneracy condition. Actually, when Z is an isolated point z, (III) becomes ker D 2 f 0 (z) = {0}, which just means that z is a nondegenerate critical point of f 0 .
If the preceding assumptions hold true, one can use the Implicit Function Theorem to find w = w(λ, z) such that Letting Z λ = {z + w(λ, z)}, it turns out that Z λ is locally diffeomorphic to Z and any critical point of f λ constrained on Z λ is a stationary point of f λ . A manifold with this property will be called a natural constraint for f λ .
We find that the behaviour of f λ on Z λ is well approximated by the behaviour of the function Γ ≡ g |Z . In particular, critical points of Γ on Z give rise to critical . for any u ∈ Z λ and |λ| small enough. The following result was proved in [2,9].
Theorem 2.1. Assume that (1)-(3) hold and that there exists a critical point z ∈ Z of Γ such that one of the following conditions holds: (i) z is nondegenerate; (ii) z is a local proper minimum or maximum; (iii) z is isolated and the local topological degree of Γ at z, deg loc (Γ , 0) = 0.
Then for |λ| small enough, the functional f λ has a critical point u λ such that u λ → z as λ → 0.
3. The variational setting. It is well known that the space and the corresponding norm is a Hilbert space and is the closure of C ∞ 0 (R N ). The energy functional associated with problem (3) is defined by f λ : where u + = max{0, u}. By (a 2 ) the functional f λ is well defined and C 2 on D 1,2 (R N ), except for the case q = 1. If q = 1, we shall consider the C 2 energy functional I λ associated with problem (3) and is defined by Solutions of problem (3) can be found as critical points of f λ and I λ , respectively, in the case 1 < q < 2 * − 1 and q = 1. Now, let us recall some well known facts. Considering the the variational problem All the minimizers of S µ are given by the functions with ε > 0. Letting Z = {z ε : ε > 0} ⊂ D 1,2 (R N ), Z is a 1-dimensional manifold of critical points of f 0 and I 0 , diffeomorphic to (0, +∞). It is worth pointing out that Z ⊂ (7) holds.
In order to apply the perturbative method to the functionals f λ and I λ , we shall check the assumptions introduced before. Firstly, we notice that by [7], we have is a Fredholm operator of index zero. Moreover arguing as in [10], we obtain the following result: By applying the abstract method in Section 2, we construct the perturbed manifold At this point, let us introduce the auxiliary function Γ : Z → R defined by setting By the well known abstract method we then have for any u = z ε + w(λ, ε) ∈ Z λ and |λ| sufficiently small, where S N 2 µ = f 0 (z ε ). Similiarly, we have for any u = z ε + w(λ, ε) ∈ Z λ and |λ| sufficiently small, where S N 2 µ = I 0 (z ε ).

4.
Proof of the main result. In this section, we shall prove Theorem 1.1. In what follows, we always assume that (a 1 )-(a 2 ) and (7) hold. For simplicity, throughout the remainder of the paper, we also denote by C > 1 a universal positive constant.
Proof. First of all, we have Secondly, according to Kelvin translation, we obtain and hance z q+1 (x) ∈ L 1 (R N ).
Proof of Theorem 1.1. We firstly consider the case q > 1. By virtue of Lemma 4.2 and Theorem 2.1, we conclude that the functional f λ has a critical point u λ = z ε + w(λ, ε) ∈ Z λ for |λ| small enough and u λ → z ε in D 1,2 (R N ) as λ → 0. Clearly u λ ≥ 0. Finally, we can apply the Harnack-type inequality in Theorem 1.1 of [11] to prove that u λ > 0. Now, we aim to focus on the case q = 1. We shall certify that u λ is positive for |λ| sufficiently small. Firstly, we aim to prove that u λ cannot change its sign. By contradiction, we assume that u λ = u λ + + u λ − with u λ + = 0 and u λ − = 0. For convenience, we denote u + = u λ + and u − = u λ − . Since u λ solves equation (3), we deduce that At this moment we define Since a(x) ∈ L N 2 (R N ) and the embedding D 1,2 (R N ) → L 2 (R N , |x| −2 dx) is continuous, the real constant C λ,µ is well defined. Moreover, for |λ| small enough, C λ,µ > 0 and C λ,µ → S µ as λ → 0. Moreover, since u ± = 0, we have for |λ| small enough. By (10) and (11) we can deduce and hence by (12) it follows that Finally, by (10) and (13), we have for |λ| small enough. On the other hand, since u λ → z ε in D 1,2 (R N ) as λ → 0, we get µ , λ → 0, which contradicts (14). This contradiction shows that u λ can not change its sign for |λ| sufficiently small. Without loss of generality, we assume that u λ ≥ 0. Then we can apply the Harnack-type inequality in [11] to prove that u λ > 0.

Anisotropic Schrödinger equations.
In this section, we shall consider the existence of positive solutions of the elliptic problem , a(x) satisfies assumptions (a 1 )-(a 2 ). Furthermore, we shall consider B λ (x) = I + λB(x), where I is the identity matrix and B(x) = {b i,j (x)} is a real, N × N matrix which satisfies (B 1 ) for any i, j = 1, ..., N b i,j ∈ C(R N ), b i,j ≥ 0, positive somewhere and lim Now, let us denote ||| · ||| the usual norm on the space of real, N × N matrices, We shall prove the following result: Theorem 5.1. Assume (a 1 ) − (a 2 ) and (B 1 ) − (B 2 ). Moreover; suppose that (7) holds. Then there exist λ > 0 and ε ∈ (0, +∞) such that for any λ ∈ R, |λ| ≤ λ, problem (15) has a positive solution u λ and A special case of (15) is the following problem: where b λ (x) = 1 + λb(x), b(x) is a real, nonnegative function on R N . An immediate corollary of Theorem 5.1 is the following result: Theorem 5.2. Assume (a 1 ) − (a 2 ) and suppose that (7) holds. Moreover, suppose that Then there exist λ > 0 and ε ∈ (0, +∞) such that for any λ ∈ R, |λ| ≤ λ, problem (16) has a positive solution u λ and As in Section 3, we distinguish the different cases 1 < q < 2 * − 1 and q = 1. If 1 < q < 2 * − 1, solutions of (15) are found as critical points of the energy functional g λ : The functional g λ is well defined and C 2 on D 1,2 (R N ).
Otherwise if q = 1, we shall consider the energy functional J λ : The functional J λ is well defined and C 2 on D 1,2 (R N ). Now, let us introduce the auxiliary function Γ : Z → R defined by setting It is not hard to check that for any u = z ε + w(λ, ε) ∈ Z λ and |λ| sufficiently small, where S N 2 µ = g 0 (z ε ).
Proof. Firstly, by (B 1 ) we deduce for any x ∈ R N , where C and C 1 are suitable positive constants. Furthermore, by the continuity of the coefficients b i,j (x) and by applying Dominated Convergence Theorem, we obtain Moreover, thanks to Lemma 4.2, it yields that Next, by Lemma 4.2 we have lim ε→+∞ Γ 2 (ε) = 0.

JING ZHANG AND SHIWANG MA
By assumption (B 2 ), we infer for ε small enough We notice that and thus α > 2β and then we infer|x| N −α |∇z(x)| 2 ∈ L 1 (R N ). As a consequence, by (18) and by applying the Dominated Convergence Theorem, we deduce that as ε → 0 + . We can conclude that Γ(ε) achieves its maximum at some point ε with ε > 0.
Proof of Theorem 5.1. We first consider the case q > 1. By Lemma 5.3 and Theorem 2.1, we infer that the functional g λ has a critical point u λ = z ε + w(λ, ε) ∈ Z λ for |λ| small enough and u λ → z ε in D 1,2 (R N ) as λ → 0. Clearly u λ ≥ 0. Moreover, by applying the Harnack type inequality in [11] we get u λ > 0. We aim to focus on the case q = 1. As before by Lemma 5.3 and Theorem 2.1, we infer that the functional J λ has a critical point u λ = z ε + w(λ, ε) ∈ Z λ for |λ| small enough. Arguing as in the proof of the Theorem 1.1 for the case q = 1, it is possible to show that u λ cannot change its sign. Therefore we can assume u λ ≥ 0. Finally, we can apply the Harnack-type inequality to prove that u λ > 0.