Existence of Multiple Solutions for a Singular Elliptic Problem with Critical Sobolev Exponent

and Applied Analysis 3 The following Hardy-Sobolev inequality is due to Caffarelli et al. 12 , which is called Caffarelli-Kohn-Nirenberg inequality. There exist constants S1, S2 > 0 such that (∫ RN |x|−bp |u|pdx )p/p∗ ≤ S1 ∫ RN |x|−ap|∇u|pdx, ∀u ∈ C∞ 0 ( R N ) , 1.8 ∫ RN |x|− a 1 |u|dx ≤ S2 ∫ RN |x|−ap|∇u|pdx, ∀u ∈ C∞ 0 ( R N ) , 1.9 where p∗ Np/ N − pd is called the Sobolev critical exponent. In the present paper, we make the following assumptions: A1 f x ∈ L1 R, g1 ⋂ Lloc R N \ {0} for 1 < r < p, where g1 |x| a 1 rσ1 , σ1 p/ p − r ; A2 f x ∈ L2 R, g2 ⋂ Lloc R N\{0} for p < r < p∗, where g2 |x|brσ2 , σ2 p∗/ p∗−r . A3 h x ∈ L R, g3 ⋂ Lloc R N \{0} for p < s < p∗, where g3 |x|μbp , μ p∗/ p∗ −s . Then, we give some basic definitions. Definition 1.1. u ∈ X is said to be a weak solution of 1.1 if for any φ ∈ C∞ 0 R there holds ∫ RN ( |x|−ap|∇u|p−2∇u · ∇φ |u| p−2uφ |x| a 1 p ) dx ∫


Introduction and Main Results
In this paper, we consider the existence of multiple solutions for the singular elliptic problem

1.3
By the variational method on Nehari manifolds 3, 4 , the author proved the existence of at least two positive solutions and the nonexistence of solutions when some certain conditions are satisfied.When p 2 and a −1, Miotto and Miyagaki in 5 considered the semilinear Dirichlet problem in infinite strip domains −Δu u λf x |u| q−1 h x |u| p−1 , x ∈ Ω, u x 0, on ∂Ω.

1.4
The authors also proved that problem 1.4 has at least two positive solutions by the methods of Nehari manifold.For other references, we refer to 6-11 and the reference therein.In fact, motivated by 1, 2, 5 , we consider the problem 1.1 .Since our problem is singular and is studied in the whole space R N , the loss of compactness of the Sobolev embedding renders a variational technique that is more delicate.By the variational method and the theory of genus, we prove that problem 1.1 has infinitely many solutions when some suitable conditions are satisfied.
In order to state our result, we introduce some weighted Sobolev spaces.For r, s ≥ 1 and g g x > 0 in R N , we define the spaces L r R N , g and L s R N , g as being the set of Lebesgue measurable functions u : R N → R 1 , which satisfy

1.5
Particularly, when g x ≡ 1, we have We denote the completion of where 1 < p < N and a < N − p /p.It is easy to find that X is a reflexive and separable Banach space with the norm u X .
The following Hardy-Sobolev inequality is due to Caffarelli et al. 12 , which is called Caffarelli-Kohn-Nirenberg inequality.There exist constants S 1 , S 2 > 0 such that where p * Np/ N − pd is called the Sobolev critical exponent.
In the present paper, we make the following assumptions: Then, we give some basic definitions.

1.10
Let I u : X → R 1 be the energy functional corresponding to problem 1.1 , which is defined as for all u ∈ X.Then the functional I ∈ C 1 X, R 1 and for all ϕ ∈ X, there holds

1.12
It is well known that the weak solutions of problem 1.1 are the critical points of the functional I u , see 13 .Thus, to prove the existence of weak solutions of 1.1 , it is sufficient to show that I u admits a sequence of critical points in X.
Our main result in this paper is the following.
Then problem 1.1 has infinitely many solutions in X.

Preliminary Results
Our proof is based on variational method.One important aspect of applying this method is to show that the functional I u satisfies PS c condition which is introduced in the following definition.
Definition 2.1.Let c ∈ R 1 and X be a Banach space.The functional I u ∈ C 1 X, R satisfies the PS c condition if for any {u n } ⊂ X such that The following embedding theorem is an extension of the classical Rellich-Kondrachov compactness theorem, see 14 .
Now we prove an embedding theorem, which is important in our paper.

Lemma 2.3. Assume
Proof.We split our proof into two cases.
By the H ölder inequality and 1.9 we have that where g 1 |x| a 1 rσ 1 , σ 1 p/ p − r .Then the embedding is continuous.Next, we will prove that the embedding is compact.Let B R be a ball center at origin with the radius R > 0. For the convenience, we denote L r R N , f by Z, that is, Z L r R N , f .Assume {u n } is a bounded sequence in X.Then {u n } is bounded in X B R .We choose α 0 in Lemma 2.2, then there exist u ∈ Z B R and a subsequence, still denoted by {u n }, such that u n − u L r B R → 0 as n → ∞.We want to prove that lim where In fact, we obtain from 2.2 that The Then 2.4 and 2.5 imply that which gives 2.3 .
In the following, we will prove that u n → u strongly in Z R N .Since X is a reflexive Banach space and {u n } is bounded in X.Then we may assume, up to a subsequence, that u n u in X.

2.7
In view of 2.3 , we get that for any ε > 0 there exists R ε > 0 large enough such that On the other hand, due to the compact embedding Therefore, there is N 0 > 0 such that for n > N 0 .Thus, the inequalities 2.8 and 2.10 show that

2.11
This shows that ii Consider p ≤ r < p * .It follows from 1.8 and the H ölder inequality that where g 2 |x| brσ 2 , σ 2 p * / p * − r .Thus, the fact of f ∈ L σ 2 R N , g 2 and 2.12 imply that the embedding is continuous.Similar to the proof of i we can also prove that the embedding Similarly, we have the following result of compact embedding.
Lemma 2.4.Assume 1 < p < s < p * and A 3 , then the embedding X → L s R N , h is compact.
The following concentration compactness principle is a weighted version of the Concentration Compactness Principle II due to Lions 15-18 , see also 19, 20 .
where μ, η are measures supported on Ω and M R N is the space of bounded measures in R N .Then there are the following results.
1 There exists some at most countable set J, a family where
Proof.We will split the proof into three steps.
Step 1. {u n } is bounded in X.
Let {u n } be a PS c sequence of I u in X, that is,

2.19
Then, we have

2.20
Since p > 1, 2.20 shows that {u n } is bounded in X.
Step 2. There exists The inequality 1.8 shows that {u n } is bounded in L p * R N , |x| −bp * .Then the above argument and the compactness embedding in Lemma 2.2 mean that the following convergence hold:

2.21
It follows from Lemma 2.5 that there exist nonnegative measures μ and η such that Thus, in order to prove For the proof of η j 0, we define the functional where x j belongs to the support of dη.It follows from 2.1 that lim n → ∞ I u n , u n ψ 0.

2.25
Since u n X is bounded, we can get from 1.8 -1.9 , Lemmas 2.3 and 2.5 that

2.26
On the other hand, where B 2ε B x j , 2ε .Then μ j η j ; furthermore, 2.16 implies that μ j η j 0 or η j > S p * / p * −p 1 .We will prove that the later does not hold.Suppose otherwise, there exists some j 0 ∈ J such that η j 0 > S p * / p * −p 1 .Then 2.19 and Lemma 2.4 show that which contradicts the hypothesis of c.Then μ j η j 0. Similarly, we define the functional

2.29
Then, the similar proof as above shows that η ∞ μ ∞ 0. Thus, we can deduce from 2.22 that Step 3. {u n } converges strongly in X.
The following inequalities 21 play an important role in our proof: Our aim is to prove that {u n } is a Cauchy sequence of X.In fact, let ψ u n − u m in 1.12 , it follows from 2.19 that

2.33
Using the inequalities 2.31 , we can get by direct computation that

2.34
with some constant c > 0, independent of n and m.
Then the H ölder inequality together with 1.8 and 2.30 yield that Similarly, we have from the H ölder inequality, Lemmas 2.3 and 2.4 that

2.36
Therefore, the above estimates imply that u n − u m X → 0 n, m → ∞ , that is, {u n } is a Cauchy sequence of X.Then {u n } converges strongly in X and we complete the proof.
Similarly, we have the following lemma., where S 1 , S 2 are as in 1.8 , and 1.9 respectively.Proof.Step 1. {u n } is bounded in X.
Let {u n } be a PS c sequence of I u in X.Then we have from Lemma 2.3 that

2.37
Since 1 < r < p < s, 2.37 shows that u n is bounded in X.
Step 2. There exists {u n } in X such that u n → u in L p * R N .Similar to the proof of Lemma 2.5, we can get that μ j η j 0 or η j > S p * / p * −p 1 by applying the functional ψ.Now we prove that there is no j 0 ∈ J such that η j 0 > S p * / p * −p 1 .Suppose otherwise, then

2.39
Then q t has the unique minimum point at

2.40
Then it follows from 2.38 that which contradicts the hypothesis of c.
Step 3. {u n } converges strongly in X.By Lemma 2.4, this result can be similarly obtained by the method in Lemma 2.6, so we omit the proof.

Existence of Infinitely Solutions
In this section, we will use the minimax procedure to prove the existence of infinity many solutions of problem 1.1 .Let A denotes the class of A ⊂ X \ {0} such that A is closed in X and symmetric with respect to the origin.For A ∈ A, we recall the genus γ A which is defined by If there is no mapping φ as above for any m ∈ N, then γ A ∞, and γ ∅ 0. The following proposition gives some main properties of the genus, see 13, 22 .Proof of Theorem 1.2.In view of Lemmas 2.6 and 2.7, I u satisfies the PS c condition in X.Furthermore, as the standard argument of 13, 22, 23 , Lemma 3.3 gives that I u has infinity many critical points with negative values.Thus, problem 1.1 has infinitely many solutions in X, and we complete the proof.

Proposition 3 . 1 .Lemma 3 . 2 . 2 Then
Let A, B ∈ A. Then 1 if there exists an odd map g ∈ C A, B , then γ A ≤ γ B ,2 if A ⊂ B, then γ A ≤ γ B , 3 γ A B ≤ γ A γ B .4 if S is a sphere centered at the origin in R N , then γ S N,5 if A is compact, then γ A < ∞ and there exists δ > 0 such that N δ A ∈ A and γ N δ A γ A , where N δ A {x ∈ X : x − A ≤ δ}.Assume (A 1 )-(A 3 ).Then for any m ∈ N, there exists ε ε m > 0 such that γ {u ∈ X : I u ≤ −ε} ≥ m.3.S σ is a sphere centered at the origin with radius of σ and S σ ⊂ {u ∈ X : I u ≤ −ε} I −ε .3.12Therefore, Proposition 3.1 shows that γ I −ε ≥ γ S σ m.Let A m {A ∈ A : γ A ≥ m}.It is easy to check that A m 1 ⊂ A m m 1, 2, . . .difficult to find that c 1 ≤ c 2 ≤ • • • ≤ c m ≤ • • • .3.14and c m > −∞ for any m ∈ N since I u is coercive and bounded below.Furthermore, we define the setK c u ∈ X : I u c, I u 0 .3.15Then, K c is compact and we have the following important lemma, see 22 .

Lemma 3 . 3 .
All the c m are critical values of I u .Moreover, if c c m c m 1 • • • c m τ , then γ K c ≥ 1 τ.
and a family {η j | j ∈ J} of positive numbers such that η |x| −bp * |u| p *