Infinitely many solutions for a singular elliptic equation involving critical Sobolev-Hardy exponents in RN

Acta Mathematica Scientia 2010,30B(3):830–840 http://actams.wipm.ac.cn

INFINITELY MANY SOLUTIONS FOR A SINGULAR ELLIPTIC EQUATION INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS IN RN ∗


)

He Xiaoming (

School of Sciences, Central University for Nationalities, Beijing 100081, China E-mail: [email protected]


)

Zou Wenming (

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China E-mail: [email protected]

Abstract In this article, we study the existence of multiple solutions for the singular semilinear elliptic equation involving critical Sobolev-Hardy exponents ∗

−Δu − μ

|u|2 (s)−2 u u =α + βa(x)|u|r−2 u, x ∈ RN . 2 |x| |x|s

By means of the concentration-compactness principle and minimax methods, we obtain infinitely many solutions which tend to zero for suitable positive parameters α, β. Key words Singular elliptic equation; Multiple solutions; Critical Sobolev-Hardy exponent; Minimax method 2000 MR Subject Classification

1

35J60; 35J25

Introduction We consider the semilinear elliptic problem ∗

−Δu − μ

|u|2 (s)−2 u u =α + βa(x)|u|r−2 u, x ∈ RN , 2 |x| |x|s

where 0 ≤ s < 2, 0 ≤ μ < μ ¯= ∗ 

 N −2 2

∗

2

, 2∗ (s) =

2(N −s) N −2

(1.1)

is the critical Sobolev-Hardy exponent, 

1 < r < 2 < 2 = 2 (0) = 2N/(N − 2), N ≥ 3, 0 ≤ a(x) ∈ Lr (RN ) with r = 2∗ /(2∗ − r), α, β are real parameters. When s = 0, α = β = 1, equation (1.1) simplifies as −Δu − μ ∗ Received

∗ u = |u|2 −2 u + a(x)|u|r−2 u, x ∈ RN . 2 |x|

October 9, 2007. Supported by NSFC (10971238, 10871109)

(1.2)

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Recently, D. Kang and Y. Deng [14] studied equation (1.2) and obtained a nontrivial solution by the Mountain Pass Lemma [19] under the following conditions: (A) a(x) is nonegative and locally bounded in RN \{0}, a(x) = O(|x|−s ) in the bounded neighborhood G of the origin, a(x) = O(|x|−t ) as |x| → ∞, 0 ≤ s < t < 2, 2∗ (t) < r < 2∗ (s), where 2∗ (β) := 2(N − β)/(N − 2) for 0 ≤ β ≤ 2; √ √ √ √ (B) r > max{(N − s)/( μ + μ − μ), (N − s − 2 μ − μ)/ μ}. We remark that there is a lot of existence results for such kind of problems in bounded domains after the celebrated paper by H. Brezis and L. Nirenberg [1]. We refer the readers to [1,2,4,5,6-9,11-13,16] in the bounded domain cases. The existence results in unbounded domains can be found in [3,10,15] and the references therein. We also refer them to [7,12,13] for the existence of positive solutions and [11] for sign-changing solutions involving Hardy-Sobolev exponents. The main purpose of this article is to establish a relationship between the multiplicity of solutions of (1.1) and the locations of the parameters α, β using Kajikiya’s Mountain Pass Lemma (Theorem 1 [18]). We prove that there exists a sequence of solutions for (1.1) with negative critical values tending to zero. Before stating the main results, we say a few words for the working space. Denote by H :=   12  u2 H(RN ) the completion of C0∞ (RN ) with respect to the norm u := RN |∇u|2 − μ |x| . 2 The scalar product in H is   uv  uv − μ 2 , ∀u, v ∈ H. (u, v) := |x| RN By the Hardy inequality [7,8], for 0 ≤ μ < μ ¯, this norm is equivalent to the usual norm 1 2 2 |∇u| . Define RN    u2 2 |∇u| − μ |x|2 RN Aμ,s = inf   2∗2(s) . u∈H\{0} 2∗ (s)



RN

|u| |x|s

Then from [12,13], we know that Aμ,s is attained by the functions 

2(¯ μ−μ)(N −s) √ μ ¯

√ μ ¯  2−s

u (x) =   > 0. √ −2  ,   N2−s (2−s) μ−μ ¯ √ √ √ μ ¯ − μ ¯ −μ μ ¯ |x|  + |x| The functions u (x) solve the equation ∗

u |u|2 (s)−2 −Δu − μ 2 = u, |x| |x|s ∗

in RN \{0},

N −s

2−s and satisfy u 22∗ = u 2 = Aμ,s . Consider the energy functional    ∗  1 |u|2 (s) α u2  β I(u) = |∇u|2 − μ 2 − ∗ − a(x)|u|r , 2 RN |x| 2 (s) RN |x|s r RN

then I(u) is well defined and I ∈ C 1 (H, R), and the critical points of I(u) correspond to the solutions of (1.1).

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Our main result can be formulated as the following.  Theorem 1.1 Assume that 1 < r < 2 and 0 ≤ a(x) ∈ Lr (RN ). Then, (i) for ∀β > 0, there exists α0 > 0 such that if 0 < α < α0 , then equation (1.1) has a sequence of solutions {un } with I(un ) ≤ 0, I(un ) → 0 and {un } converges to zero as n → ∞; (ii) for ∀α > 0, there exists β0 > 0 such that if 0 < β < β0 , then equation (1.1) has a sequence of solutions {un } with I(un ) ≤ 0, I(un ) → 0 and {un } converges to zero as n → ∞. The rest of the article is organized as follows. In Section 2, we give some preliminary results including a slight variant of concentration compactness lemma from D. Smets [17] and M. Willem [20], and deal with the compactness properties of I(u) in H. In Section 3, we give the proof of Theorem 1.1.

2

Preliminaries

Throughout this article, we denote by M (RN ∪ {∞}) (respectively, M + (RN ∪ {∞}) the space of finite Radon measures (respectively, positive finite Radon measures) on RN ∪ {∞}. We   write γn  γ in M (RN ∪ {∞}) if RN f dγn → RN f dγ for all f ∈ C(RN , R). · p is the norm in Lp (RN ) as usual. δx is the Dirac mass at point x and c is the generic positive constant. We need to apply a suitable concentration-compactness principle, refer to [5] for the original version. The space RN ∪ {∞} is given the standard topology that makes it compact, which means that the measures can be identified as the dual space C(RN ∪ {∞}). For example, δ∞ is well defined and δ∞ (ϕ) = ϕ(∞).  Proposition 2.1 The function F (u) := RN a(x)|u|r is well defined and weakly continuous on H. Moreover, F is continuously differentiable, its derivative F  : H → H ∗ is given by 

F  (u), v = r RN a(x)|u|r−2 uv, and maps weakly convergent sequence to weakly convergent ones. Proposition 2.2 Let 0 ≤ s < 2, {un } ⊂ H be a bounded sequence, going if necessary to a subsequence, still denoted by {un },there exist u ∈ H, η, ν, ν¯p ∈ M (RN ∪ {∞}), an at most countable set J and a set of different points {xj }i∈J ⊂ RN \{0} such that un  u weakly in H, un → u strongly in Lloc (RN ), 1 ≤ r < 2∗ , and un → u a.e. in RN , and furthermore, (c1) |∇un |2  η ≥ |∇u|2 + η0,s δ0 + η∞,s δ∞ , if 0 < s < 2, where η0,s ≥ 0, η∞,s ≥ 0;  |∇un |2  η ≥ |∇u|2 + η0,0 δ0 + η∞,0 δ∞ + ηj δxj , if s = 0, where η0,0 ≥ 0, η∞,0 ≥ 0, ηj ≥ 0, 0 < |xj | < ∞, ∀j ∈ J ; 2∗ (s)

j∈J

2∗ (s)

| (c2) |un|x|  ν = |u||x|s + ν0,s δ0 + ν∞,s δ∞ , if 0 < s < 2, where ν0,s ≥ 0, ν∞,s ≥ 0; s  ∗ ∗ |un |2  ν = |u|2 + ν0,0 δ0 + ν∞,0 δ∞ + νj δxj , if s = 0, where ν0,0 ≥ 0, ν∞,0 ≥ 0, νj ≥ 0, j∈J

0 < |xj | < ∞, ∀j ∈ J ; 2 u2 (c3) |u|x|n 2|  ν¯2 = |x| 2 + ν0,2 δ0 + ν∞,2 δ∞ , where ν0,2 ≥ 0, ν∞,2 ≥ 0. Here, we have used the following quantities:  |∇un |2 , 0 < s < 2; η∞,s = lim lim sup R→∞

n→∞

|x|>R

 η∞,0 = lim lim sup R→∞

n→∞

|x|>R

|∇un |2 ,

s = 0;

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 ν∞,s = lim lim sup R→∞

n→∞

∗

|x|>R



ν∞,0 = lim lim sup R→∞

n→∞

|un |2 (s) , |x|s ∗

|x|>R

n→∞

0 < s < 2;

|un |2 ,



ν∞,p = lim lim sup R→∞

833

|x|>R

0 = s;

|un |2 . |x|2

The proof of Proposition 2.1 is similar to that of Lemma 3.10 [20], and the proof of Proposition 2.2 is essential to Lemma 1.40 [20] and [17]. We recall that a C 1 functional I on Banach space X is said to satisfy the Palais-Smale condition at level c ((P S)c in short) if every sequence {un } ⊂ X satisfying lim I(un ) = c and n→∞

lim I  (un ) X ∗ = 0 has a convergent subsequence. n→∞ Proposition 2.3 Let 1 < r < 2, 0 ≤ s < 2. Then, (i) for ∀β > 0, there exists α0 > 0, such that, if 0 < α < α0 and c < 0, then, I(u) satisfies (P S)c ; (ii) for ∀α > 0, there exists β0 > 0, such that, if 0 < β < β0 and c < 0, then, I(u) satisfies (P S)c . Proof Let {un } be a sequence in H, such that I(un ) → c < 0 and I  (un ) → 0,

(2.1)

then, for n large enough, by H¨ older inequality and the definition of Aμ,0 , we have 1

I  (un ), un  2∗ (s)    1 1 1  1  u2n  2 − − ∗ |∇un | − μ 2 − β a(x)|un |r = 2 2∗ (s) RN |x| r 2 (s) RN 1 1  − r2 2−s un 2 − β − ∗ A a r un r , ≥ 2(N − s) r 2 (s) μ,0

o(1)(1 + un ) + |c| ≥ I(un ) −

∗

where r = 2∗2−r . It follows that {un } is bounded in H. Due to Hardy-Sobolev inequality, we can assume, going if necessary to a subsequence, still denoted by {un }, that un  u in H, un → u a.e. in RN , and satisfies (c1)–(c3) of Proposition 2.2. Let xj be a singular point of the measures μ and ν, we define a function φ ⊂ C0∞ (RN , [0, 1]), such that φ(x) = 1 in B(xj , ); φ(x) = 0 in RN \ B(xj , 2) and |∇φ| ≤ 2/ in RN , where B(xj , ) is a ball in H centered at xj with radius . Then, it is seen that {φun } is bounded in H. It follows from (2.1) that I  (un ), φun  → 0, that is,    u2  |∇un |2 φ − μ n2 φ − lim un ∇un ∇φ = lim n→∞ RN n→∞ |x| RN

  ∗ −α |x|−s |un |2 (s) φ − β a(x)|un |r φ . (2.2) RN

Furthermore, (c1)–(c3) imply that   |∇un |2 φ = lim n→∞

RN

 lim

n→∞

RN

∗

|un |2 φ =



RN

 RN

RN

φdη ≥

RN

 φdν =

RN

|∇u|2 φ + ηj , ∗

|u|2 φ + νj ,

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  lim lim  ε→0 n→∞ 

   12    12  un ∇un ∇φ ≤ lim lim |un |2 |∇φ|2 |∇un |2 ε→0 n→∞ ¯N R RN RN  12  ≤ c lim |u|2 |∇φ|2 ε→0 RN   N1    21∗ ∗ ≤ c lim |∇φ|N |u|2 ε→0

≤ c lim

B(xj ,2ε)



ε→0

  lim lim  ε→0 n→∞

RN

B(xj ,2ε)

 21∗ ∗

|u|2

B(xj ,2ε)

= 0,

     |un |2  |un |2   = 0, φ ≤ lim lim φ    2 2 ε→0 n→∞ |x| B(xj ,2ε) (|xj | − 2ε)  a(x)|un |r φ = 0. lim lim ε→0 n→∞

RN

From the above augument, it follows that 0 = lim lim I  (un ), un φ ≥ ηj − ανj . ε→0 n→∞ 2

By Sobolev inequality, we get A0,0 (νj ) 2∗ ≤ ηj , thus, (i) νj = 0

p

(ii) νj ≥ (α−1 A0,0 ) 2 ,

or

which implies that J is finite. To obtain information about possible concentration at infinity, we define a cut off function ψ ∈ C0∞ (RN , [0, 1]), such that ψ(x) = 0 on |x| ≤ 1 and ψ(x) = 1 on |x| ≥ 2 and we set ψR (x) = ψ(x/R). Then, {ψR un } is bounded in H, and lim I  (un ), ψR un  = 0, that is, − lim

n→∞

n→∞



 RN

un ∇un ∇ψR = lim

n→∞



−α

RN



 u2n ψR 2 |x| RN

 −s 2∗ (s) r |x| |un | ψR − β a(x)|un | ψR . |∇un |2 ψR − μ

(2.3)

RN

By the definition of Aμ,s , we have  ∗   |un ψR |2 (s)  2∗2(s) |un ψR |2  |∇(un ψR )|2 − μ ≥ A . μ,s |x|2 |x|s RN RN Consequently,  μ  ≤

RN

RN

 =

RN

 |un ψR |2 + A μ,s |x|2



(s) 

∗

RN

|un ψR |2 |x|s

|ψR ∇un + un ∇ψR |2   2 |∇un |2 ψR + u2n |∇ψR |2 + 2 RN

RN

2 2∗ (s)

∇un ∇ψR · un ψR .

(2.4)

Furthermore, by H¨ older inequality we have  RN

|ψR ∇un | · |un ∇ψR | ≤

 R<|x|<2R

|un ∇ψR |2

 12   R<|x|<2R

|∇un |2

 12

,

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which together with the boundedness of {un } in H implies that   12  lim sup |ψR ∇un | · |un ∇ψR | ≤ c lim sup |un |2 |∇ψR |2 n→∞

RN

 =c ≤c ≤c

Thus,

 

n→∞

R<|x|<2R

R<|x|<2R

|u|2 |∇ψR |2 2∗

R<|x|<2R

|u|

∗

R<|x|<2R

|u|2

 lim lim sup

R→∞

n→∞

RN

|ψR ∇un | · |un ∇ψR | ≤ lim c

Similarly, we obtain lim lim sup R→∞

RN

n→∞

 21∗    21∗

R<|x|<2R

|∇ψR |N

 N1

.



R→∞



 12

∗

R<|x|<2R

|u|2

 21∗

= 0.

u2n |∇ψR |2 = 0. Therefore, we see from (2.4) that 2

η∞,s − μν∞,p ≥ Aμ,s (ν∞,s ) 2∗ (s) , 0 ≤ s < 2,

(2.5)

and by H¨ older inequality and the definition of Aμ,0 ,  |

r

RN

a(x)|un | ψR | ≤



2∗

|x|≥R

|un |

− r2

≤ (Aμ,0 )

un

 2r∗  

r

|x|≥R



|x|≥R

|a(x)|

|a(x)|

2∗ 2∗ −r

2∗ 2∗ −r

 2∗2−r ∗

 2∗2−r ∗

,

which implies   lim lim sup  R→∞ n→∞

    a(x)|un | ψR  ≤ c lim R→∞ r

RN

|x|≥R

|a(x)|

2∗ 2∗ −r

 2∗2−r ∗

= 0.

Therefore, take limit by letting R → ∞ in (2.3), we have η∞,s − μν∞,2 ≤ αν∞,s , 0 ≤ s < 2.

(2.6)

By (2.5) and (2.6), we conclude (iii) ν∞,s = 0 or

N −s

(iv) ν∞,s ≥ (α−1 Aμ,s ) 2−s ,

0 ≤ s < 2.

In contrast, Hardy inequality implies that 0≤μ ¯ν∞,2 ≤ η∞,s ,

0 ≤ (1 − μ/¯ μ)η∞,s ≤ η∞,s − μν∞,2 .

If ν∞,s = 0, from (2.6) it follows that η∞,s = ν∞,2 = 0. The same conclusion holds for the concentration at the origin 0, namely, 2

η0,s − μν0,2 ≥ Aμ,s (ν0,s ) 2∗ (s) , 0 ≤ s < 2, and (v) ν0,s = 0

or

N −s

(vi) ν0,s ≥ (α−1 Aμ,s ) 2−s , 0 ≤ s < 2.

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Furthermore, 0≤μ ¯ ν0,2 ≤ η0,s , 0 ≤ (1 − μ/¯ μ)η0,s ≤ η0,s − ν0,2 . If ν0,s = 0, then, η0,s = ν0,2 = 0. We claim that (ii), (iv), and (vi) are impossible if α0 or β0 is chosen small enough. For this, from the lower semi-continuity of the norm and the weak continuity of F , we have   1 0 > c = lim I(un ) − ∗ I  (un ), un ) n→∞ 2 (s)

   1 1  2−s u2n  2 r − ∗ = lim |∇un | − μ 2 − β a(x)|un | n→∞ 2(N − s) RN |x| r 2 (s) RN   1 2−s 1  u2  ≥ − ∗ a r u r2∗ |∇u|2 − μ 2 − β 2(N − s) RN |x| r 2 (s) 1 1  2−s Aμ,0 u 22∗ − β − ∗ a r u r2∗ . ≥ (2.7) 2(N − s) r 2 (s) This yields that r

u r2∗ ≤ cβ 2−r .

(2.8)

If (iv) occurs, we obtain by (2.7), (2.8), and (2.5),   1 0 > c = lim I(un ) − ∗ I  (un ), un  n→∞ 2 (s)   1 1  u2  2−s − ∗ a r u r2∗ |∇un |2 − μ n2 ψR − β ≥ lim n→∞ 2(N − s) RN |x| r 2 (s) 2 2−s (η∞,s − μν∞,2 ) − cβ 2−r ≥ 2(N − s)   2∗2(s) N −2 2 2−s Aμ,s (α−1 Aμ,s ) 2−s ≥ − cβ 2−r 2(N − s) N −s N −2 2 2−s 2−s Aμ,s = α− 2−s − cβ 2−r . 2(N − s) However, if β > 0 is given, we can choose α0 > 0 so small that for every 0 < α < α0 , the last term on the right-hand side above is greater than 0, which yields a contradiction. Similarly, if α > 0 is given, we can choose β0 > 0 so small that for every 0 < β < β0 the last term on the right-hand side above is greater than 0. By a similar argument, we can prove that (ii) and (vi) cannot occur. Up to now, we have shown that   ∗ ∗ lim |un |2 (s) |x|−s = |u|2 (s) |x|−s . n→∞

RN

RN

In view of un  u in H and the Brezis-Lieb lemma [16], we have  ∗ lim |x|−s |un − u|2 (s) = 0. n→∞

RN

We are now in a position to show that {un } strongly converges to u in H. First, we have un − u 2 = I  (un ) − I  (u), un − u  ∗ |x|−s (|un |2 (s)−2 un − |u|2∗(s)−2 u)(un − u) +α N R +β a(x)(|un |r−2 un − |u|r−2 u)(un − u). RN

(2.9)

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Therefore,  un − u ≤ c | I  (un ) − I  (u), un − u|     ∗ |x|−s (|un |2 (s)−2 un − |u|2∗(s)−2 u)(un − u) +α    +β 

RN

RN

Because

r−2

a(x)(|un |

r−2

un − |u|

  12  u)(un − u) .

    −s 2∗ (s)−2 2∗ (s)−2  |x| (|un | un − |u| u)(un − u) A :=  N R  ∗ ∗ ≤ |x|−s |un |2 (s)−1 |un − u| + |x|−s |u|2 (s)−1 |un − u| RN

RN

:= A1 + A2 ,  ∗1    −s 2∗ (s) 2 (s) A1 ≤ |x| |un − u| RN

RN

∗

|x|−s |un |2

(s)

 2∗(s)−1 2∗ (s)

.

It follows from Hardy-Sobolev inequality, the boundedness of {un } in H and (2.9) that A1 → 0 as n → ∞. Similarly, A2 → 0 as n → ∞. In view of the weak continuity of F , and un  u in H, we deduce un − u → 0 as n → ∞. The proof is completed.

3

Existence of Infinitely Many Solutions

Let X be a Banach space and A a subset of X. A is said to be symmetric if u ∈ A implies −u ∈ A. Let Γ denote the family of closed symmetric subsets A of X, such that 0∈A. that is, Γ = {A ⊂ X \ {0}; A is closed in X and symmetric with respect to the origin}. For A ∈ Γ, we define γ(A) := inf{m ∈ N ; ∃ϕ ∈ C(A, Rm \{0}), ϕ(x) = −ϕ(−x)}; and Γk := {A ∈ Γ; γ(A) ≥ k}, if there is no mapping ϕ as above for any m ∈ N, then γ(A) = ∞. Moreover, we set γ(∅) = 0. We list the following main properties of the genus (cf. [19]). Proposition 3.1 Let A, B ∈ Γ. Then, (1) If there exists an odd map f ∈ 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 Rm , then γ(S) = m. (5) If A is compact, then γ(A) < ∞ and there exists δ > 0 such that Nδ (A) ∈ Γ and γ(Nδ (A)) = γ(A), where Nδ (A) = {x ∈ X; x − A ≤ δ}. We recall the following new version of symmetric mountain pass lemma due to R. Kajikiya ([Theorem 1, [18]). Lemma 3.1 Let E be an infinite dimensional Banach space and I ∈ C 1 (E, R) and suppose the following conditions hold:

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(A1) I(u) if even, bounded from below, I(0) = 0 and I(un ) satisfies the Palais-Smale condition (PS in short). (A2) For each k ∈ N, there exists an Ak ∈ Γk , such that supu∈Ak I(u) < 0. Then, (i) or (ii) below holds. (i) There exists a sequence {un } such that I  (uk ) = 0, I  (uk ) < 0 and {un } converges to zero. (ii) There exist two sequences {uk } and {vk }, such that I  (uk ) = 0, I(uk ) = 0, uk = 0, lim uk = 0; I  (vk ) = 0, I(vk ) < 0, lim I(vk ) = 0 and {vk } converges to a non-zero limit. k→∞

k→∞

Remark 3.1 From Lemma 3.1, we have a sequence {uk } of critical points such that I(uk ) ≤ 0, uk = 0 and lim un = 0. k→∞

Let I(u) be the functional defined as before, 1 < r < 2, and α > 0, β > 0. Then,     α u2  β 1 |∇u|2 − μ 2 − ∗ |x|−s |u|2∗(s) − a(x)|u|r I(u) = 2 RN |x| 2 (s) RN r RN ∗ 1 ≥ u 2 − αc1 u 2 (s) − βc2 u r , 2 where c1 , c2 are some positive constants. Using the same idea as in [2], we define Q(t) :=

∗ 1 2 t − αc1 t2 (s) − βc2 tr , 2

then, it is easy to see that, given β > 0 there exists α1 > 0 so small that for every 0 < α < α1 , there exists 0 < t0 < t1 , Q(t) > 0 for t0 < t < t1 , Q(t) < 0 for t > t1 and 0 < t < t0 . Similarly, given α > 0, we can choose β1 > 0 with the property that t0 , t1 as above exist for each 0 < β < β1 . Clearly, Q(t0 ) = 0 = Q(t1 ). Following the same idea as in [2], we consider the truncated functional     ∗ 1 α u2  β ˜ I(u) = |∇u|2 − μ 2 − ∗ ψ(u) |x|−s |u|2 (s) − a(x)|u|r , 2 RN |x| 2 (s) r N N R R where ψ(u) = τ ( u ) and τ : R+ → [0, 1] is a nonincreasing C∞ function such that τ (t) = 1 if  t ≤ t0 and τ (t) = 0 if t ≥ t1 . Obviously, I(u) is even. Thus, it follows from Proposition 2.3. Proposition 3.2 ˜ ˜ (1) If I(u) < 0, then u < t0 and I(u) = I(u). ∗ (2) For any α > 0, there exists β , such that if 0 < β < β ∗ and c < 0, then, I˜ satisfies (P S)c condition. (3) For any β > 0, there exists α∗ , such that if 0 < α < α∗ and c < 0, then, I˜ satisfies (P S)c condition. ˜ Remark 3.2 Denote Kc = {u ∈ H; I˜ (u) = 0, I(u) = c}. If α, β are as in (2) or (3) above, then, it follows (P S)c that Kc (c < 0) is compact. ˜ Proposition 3.3 Denote I˜c := {u ∈ H; I(u) ≤ c}. Then, for ∀m ∈ N, ∃m < 0, such that γ(I˜m ) ≥ m. Proof Let Hm be an m-dimensional subspace of H, for ∀u ∈ Hm , u = 0 write u = rm v with v ∈ Hm , v = 1 and rm = u . From the assumptions of a(x), it is easy to see that, for every v ∈ Hm with v = 1, there exists dm > 0, such that  a(x)|v|r ≥ dm . RN

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Therefore, for 0 < rm < t0 , we have     α u2  β 1 2 −s 2∗ (s) ˜ |∇u| − μ 2 − ∗ |x| |u| − a(x)|u|r I(u) = I(u) = 2 RN |x| 2 (s) RN r RN ∗ 1 r ≤ u 2 − αc1 u 2 (s) − βc2 dm rm 2 := m . ˜ Hence, we can choose rm ∈ (0, t0 ) so small that I(u) ≤ m < 0. Let Srm = {u ∈ H; u = rm },  then Srm ∩ Hm ⊂ I˜ k . Hence, by Proposition 3.1, γ(I˜m ) ≥ γ(Srm ∩ Hm ) = m. According to Proposition 3.2, we can denote Γm = {A ∈ Γ; γ(A) ≥ m} and let ˜ cm := inf sup I(u).

(3.1)

−∞ < cm ≤ m < 0, m ∈ N,

(3.2)

A∈Γm u∈A

Then, because I˜m ∈ Γm and I˜ is bounded from below. Proposition 3.4 Let α, β be as in (2) or (3) of Proposition 3.2. Then, all cm (given by (3.1)) are critical values of I˜ and cm → 0. Proof It is clear that cm ≤ cm+1 . By (3.2) cm < 0. Hence, cm → c∞ ≤ 0. Moreover, as ˜ (P S)c is satisfied, it follows from a standard argument [19] that all cm are critical values of I. We claim that c∞ = 0. If c∞ < 0, then by Remark 3.3 Kc∞ is compact and Kc∞ ∈ Γ. It follows from Proposition 3.1 that γ(Kc∞ ) = k < +∞ and there exists δ > 0, such that γ(Kc∞ ) = γ(Nδ (Kc∞ )) = k. By the deformation lemma ([19], Theorem A.4), there exist  > 0( + c∞ < 0) and an odd homeomorphism η : H → H, such that η(I˜c∞ + \Nδ (Kc∞ )) ⊂ I˜c∞ − .

(3.3)

Because cm is increasing and converges to c∞ , there exists m ∈ N , such that cm > c∞ −  and ˜ cm+k ≤ c∞ . Choose A ∈ Γm+k , such that sup I(u) < c∞ + , that is, u∈A

A ⊂ I˜c∞ + . Now, it follows from Proposition 3.1 that γ(A \ Nδ (Kc∞ )) ≥ γ(A) − γ(Nδ (Kc∞ ) ≥ m, γ(η(A \ Nδ (Kc∞ ))) ≥ m. Therefore, η(A \ Nδ (Kc∞ )) ∈ Γm , consequently, sup

˜ I(u) ≥ cm > c∞ − .

u∈η(A\Nδ (Kc∞ ))

In contrast, by (3.3) and (3.4), η(A \ Nδ (Kc∞ )) ⊂ η(I˜c∞ + \ Nδ (Kc∞ )) ⊂ I˜c∞ − .

(3.4)

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ACTA MATHEMATICA SCIENTIA

Vol.30 Ser.B

This yields a contradiction. Hence, cm → 0.   Proof of Theorem 1.1 By (i) of Proposition 3.2, I(u) = I(u) if I(u) < 0. For each k k   k ∈ N, put Ak = I (= I ). In addition, I is even and bounded from below. Then, by propositions 3.2–3.4, one can see that all the assumptions of Lemma 3.1 are satisfied, and the proof is completed. Remark 3.3 In [14], the authors obtained only one nontrivial solution for (1.2) under the assumptions (A) and (B) where a(x) is too restrictive. In this article, we consider problem (1.1) which is more general than (1.2) and get infinitely many solutions. Hence, we have extended and improved the main result of [14]. References [1] Brezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical exponents. Comm Pure Appl Math, 1983, 34: 437–477 [2] Azorero J G, Alonso I P. Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term. Trans Amer Math Soc, 1991, 323: 877–895 [3] Ben-Naoum A K, Troestler C, Willem M. Extrema problems with critical Sobolev exponents on unbounded dommains. Nonlinear Analysis, 1996, 26: 823–833 [4] Bianchi G, Chabrowski J, Szulkin A. On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent. Nonlinear Analysis, 1995, 25: 41–59 [5] Lions P L. The concentration-compactness principle in the caculus of variation. The limit case. Revista Mat. Iberoamer, 1985, 1: 45–120; 145–201 [6] Jannelli E. The role played by space dimension in elliptic critical problems. J Differential Equations, 1999, 156: 407–426 [7] Ferrero A, Gazzola F. Existence of solutions for singular critical growth semilinear elliptic equations. J Differntial Equations, 2001, 177: 494–522 [8] Ghoussoub N, Yuan C G. Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponets. Trans Amer Math Soc, 2000, 352: 5703–5743 [9] Li S J, Zou W M. Remarks on a class of elliptic problems with critical exponents. Nonlinear Analysis, 1998, 32: 769–774 [10] Zou W M. On finding sign-changing solutions. J Functional Analysis, 2006, 234: 364–419 [11] Cao D M, Peng S J. A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms. J Differential Equations, 2003, 193: 424–434 [12] Kang D S, Peng S J. Existence of solutions for elliptic problems with critical Sobolev-Hardy exponents. Israel J Mathematics, 2004, 143: 281–297 [13] Kang D S, Peng S J. Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy potential. Appl Math Lett, 2005, 15: 1094–1100 [14] Kang D S, Deng Y B. Existence of solutions for a singular critical elliptic equation. J Math Anal Appl, 2003, 284: 724–732 [15] Ruiz D, Willem M. Ellipltic problems with critical exponents and Hardy potentials. J Differential Equations, 2003, 190: 524–538 [16] Brezis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88: 486–490 [17] Smets D. A concentration-compactness principle lemma with applications to singular eigenvalue problems. J Functional Analysis, 1999, 167: 463–480 [18] Kajikiya R. A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J Functional Analysis, 2005, 225: 352–370 [19] Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBME Regional Conf Ser in Math, No.65, Amer Math Soc, Providence RI, 1986 [20] Willem M. Minimax Theorems. Basel: Birkh¨ auser, 1996