Open Access

Existence of multiple solutions for the Brezis-Nirenberg-type problem with singular coefficients

Boundary Value Problems20122012:137

DOI: 10.1186/1687-2770-2012-137

Received: 4 September 2012

Accepted: 19 September 2012

Published: 26 November 2012

Abstract

By energy estimates and by establishing a local (PS) condition, we obtain the multiplicity of solutions to a class of Brezis-Nirenberg-type problem with singular coefficients via minimax methods and the Krasnoselskii genus theory.

Keywords

Brezis-Nirenberg-type problem minimax method

1 Introduction and main results

This paper is concerned with multiple solutions for the semilinear Brezis-Nirenberg-type problem with singular coefficients
{ div ( D u | x | 2 a ) = λ | u | 2 2 u | x | 2 b + β | u | q 2 u | x | α , x Ω ; u = 0 , x Ω ,
(1)

where Ω R n is a bounded smooth domain, and 0 Ω , < a < n 2 2 , a b < a + 1 , 2 = 2 n n 2 d , d = a + 1 b ( 0 , 1 ] , 1 < q < 2 , α < ( 1 + a ) q + n ( 1 q 2 ) . β > 0 , λ > 0 are two real parameters.

The starting point of the variational approach to the problem is the Caffarelli-Kohn-Nirenberg inequality (see [1]): There is a constant C a , b > 0 such that
( R n | x | 2 b | u | 2 d x ) 2 / 2 C a , b R n | x | 2 a | D u | 2 d x ,
(2)
for all u C 0 ( R n ) , where
< a < n 2 2 , a b < a + 1 , 2 = 2 n n 2 d , d = a + 1 b .
Let D a 1 , 2 ( Ω ) be the completion of C 0 ( R n ) with respect to the weighted norm defined by
u = ( Ω | x | 2 a | D u | 2 d x ) 1 / 2 .
From the boundedness of Ω and the standard approximation arguments, it is easy to see that (2) holds for any u D a 1 , 2 ( Ω ) in the sense:
( Ω | x | α | u | r d x ) 2 / r C Ω | x | 2 a | D u | 2 d x
(3)
for 1 r 2 = 2 n n 2 , α r ( 1 + a ) + n ( 1 r 1 2 ) , that is, the embedding D a 1 , 2 ( Ω ) L r ( Ω , | x | α ) is continuous, where L r ( Ω , | x | α ) is the weighted L r space with the norm
u r , α : = u L r ( Ω , | x | α ) = ( Ω | x | α | u | r d x ) 1 / r .
On D a 1 , 2 ( Ω ) , we can define the energy functional
J ( u ) = 1 2 Ω | x | 2 a | D u | 2 d x λ 2 Ω | x | 2 b | u | 2 d x β q Ω | x | α | u | q d x .
(4)

From (4), J is well defined in D a 1 , 2 ( Ω ) , and J C 1 ( D a 1 , 2 ( Ω ) , R ) . Furthermore, the critical points of J are weak solutions of problem (1).

Breiz-Nirenberg-type problems have been generalized to many situations such as
{ div ( D u | x | 2 a ) μ u | x | 2 ( a + 1 ) = | u | 2 2 u | x | 2 b + λ u | x | 2 ( a + 1 ) c , x Ω , u = 0 , x Ω .
(5)

Xuan et al.[2] derived the explicit formula for the extremal functions of the best embedding constant by applying the Bliss lemma [3]. They got a nontrivial solution for problem (5) including the resonant and nonresonant cases by variational methods. He and Zou [4] studied problem (5) and obtained the multiplicity of solutions with the aid of a pseudo-index theory. In [5], problem (5) has been extended to the p-Laplace case by Xuan.

The purpose of this paper is to study the multiplicity of solutions for the Breiz-Nirenberg-type problem (1) with the aid of a minimax method. We obtain multiple nontrivial solutions of (1) by proving the local (PS) condition and energy estimates.

Our main results are the following.

Theorem 1.1 Suppose 1 < q < 2 , then
  1. (i)

    β > 0 , λ 0 > 0 such that if 0 < λ < λ 0 , problem (1) has a sequence of solutions { u m } with J ( u m ) < 0 and J ( u m ) 0 as m .

     
  2. (ii)

    λ > 0 , β 0 > 0 such that if 0 < β < β 0 , problem (1) has a sequence of solutions { u m } with J ( u m ) < 0 and J ( u m ) 0 as m .

     

2 Preliminary results

Lemma 2.1[5]

Suppose that Ω R n is an open bounded domain with C 1 boundary and 0 Ω , < a < ( n 2 ) / 2 . The embedding D a 1 , 2 ( Ω ) L r ( Ω , | x | α ) is compact if 1 r < 2 , α < ( 1 + a ) r + n ( 1 r 2 ) .

Lemma 2.2 (Concentration compactness principle [5])

Let < a < ( n 2 ) / 2 , a b a + 1 , 2 = 2 n / ( n 2 d ) , d = 1 + a b [ 0 , 1 ] , and M ( R n ) be the space of bounded measures on R n . Suppose that { u m } D a 1 , 2 ( R n ) is a sequence such that
u m u in  D a 1 , 2 ( R n ) , μ m : = | | x | a D u m | 2 d x μ in  M ( R n ) , ν m : = | | x | b u m | 2 d x ν in  M ( R n ) , u m u a.e. on  R n .
Then there are the following statements:
  1. (1)
    There exists some at most countable set I, a family { x ( i ) : i I } of distinct points in R n , and a family { ν ( i ) : i I } of positive numbers such that
    ν = | | x | b u | 2 d x + i I ν ( i ) δ x ( i ) ,
    (6)
     
where δ x is the Dirac-mass of mass 1 concentrated at x R n .
  1. (2)
    The following inequality holds
    μ | | x | a D u | 2 d x + i I μ ( i ) δ x ( i )
    (7)
     
for some family { μ ( i ) > 0 : i I } satisfying
S ( ν ( i ) ) 2 / 2 μ ( i ) for all  i I ,
(8)
where S : = inf u D a 1 , 2 ( R n ) { 0 } E a , b ( u ) to be the best embedding constants, and
E a , b ( u ) = R n | x | 2 a | D u | 2 d x ( R n | x | 2 b | u | 2 d x ) 2 / 2 .

In particular, i I ( ν ( i ) ) 2 / 2 < .

Lemma 2.3 Assume { u n } is a (PS) c sequence with c < 0 , 1 < q < 2 , then
  1. (1)

    λ > 0 , there exists β 1 > 0 such that for any 0 < β < β 1 , { u n } has a convergent subsequence in D a 1 , 2 ( Ω ) .

     
  2. (2)

    β > 0 , there exists λ 1 > 0 such that for any 0 < λ < λ 1 , { u n } has a convergent subsequence in D a 1 , 2 ( Ω ) .

     

Proof (1) The boundedness of (PS) c sequence.

For { u n } is a (PS) c sequence, then
J ( u n ) = 1 2 Ω | x | 2 a | D u n | 2 d x λ 2 Ω | x | 2 b | u n | 2 d x β q Ω | x | α | u n | q d x ,
(9)
J ( u n ) , u n = Ω | x | 2 a | D u n | 2 d x λ Ω | x | 2 b | u n | 2 d x β Ω | x | α | u n | q d x .
(10)
So, we get
o ( 1 ) ( 1 + u n ) + | c | J ( u n ) 1 2 J ( u n ) , u n = d n u n 2 ( 1 q 1 2 ) β Ω | u n | q | x | α d x d n u n 2 ( 1 q 1 2 ) β C α u n q .
We have the boundedness of { u n } for 1 < q < 2 , then there exists a subsequence, we still denote it by { u n } , such that
u n u in  D a 1 , 2 ( Ω ) , u n u in  L 2 ( Ω , | x | 2 b ) , u n u in  L r ( Ω , | x | α ) , 1 r < 2 n n 2 , α < ( 1 + a ) r + n ( 1 r 2 ) , u n u a.e. on  Ω .
From the concentration compactness principle, there exist nonnegative measures μ, ν and a countable family { x i } Ω such that
| x | 2 b | u n | 2 d x ν = | x | 2 b | u | 2 d x + i I ν ( i ) δ x ( i ) , | x | 2 a | D u n | 2 d x μ | x | 2 a | D u | 2 d x + S i I ( ν ( i ) ) 2 / 2 δ x ( i ) .
  1. (2)

    Up to a subsequence, u n u in L 2 ( Ω , | x | 2 b ) .

     
Since { u n } is bounded in D a 1 , 2 ( Ω ) , we may suppose, without loss of generality, that there exists T ( L 2 ( Ω , | x | 2 a ) ) n such that
D u n T in  ( L 2 ( Ω , | x | 2 a ) ) n .
On the other hand, | u n | 2 2 u n is also bounded in L 2 ( Ω , | x | 2 b ) and
| u n | 2 2 u n | u | 2 2 u in  L 2 ( Ω , | x | 2 b ) .
Note that
o ( 1 ) φ = J ( u n ) , φ = Ω | x | 2 a D u n D φ d x λ Ω | x | 2 b | u n | 2 2 u n φ d x β Ω | x | α | u n | q 2 u n φ d x ,
(11)
taking n in (11), we have
Ω | x | 2 a T D φ d x = λ Ω | x | 2 b | u | 2 2 u φ d x + β Ω | x | α | u | q 2 u φ d x
(12)
for any φ D a 1 , 2 ( Ω ) . Let φ = ψ u n in (12), where ψ C ( Ω ¯ ) , then it follows that
Ω | x | 2 a D u n u n D ψ d x + Ω | x | 2 a | D u n | 2 ψ d x = λ Ω | x | 2 b | u n | 2 ψ d x + β Ω | x | α | u n | q ψ d x .
(13)
Taking n in (13), we have
Ω | x | 2 a u T D ψ d x + Ω ψ d μ = λ Ω ψ d ν + β Ω | x | α | u | q ψ d x .
(14)
Let φ = ψ u in (12), then it follows that
Ω | x | 2 a T ψ u d x + Ω | x | 2 a T u d ψ = λ Ω | x | 2 b | u | 2 ψ d x + β Ω | x | α | u | q ψ d x .
(15)
Thus, it implies that
Ω ψ d μ = λ i I ν i ψ ( x i ) + Ω | x | 2 a T D u ψ d x ,
(16)
which implies that
S ( ν i ) 2 / 2 μ i = λ ν i .

Hence, ν i ( λ 1 S ) n / 2 d if ν i 0 .

On the other hand,
0 > c = lim n ( J ( u n ) 1 2 J ( u n ) , u n ) = lim n ( d n u n 2 β ( 1 q 1 2 ) Ω | x | α | u n | q d x ) d n u 2 β C u q ,
then u q C β q / ( 2 q ) , so that
0 > c = lim n ( J ( u n ) 1 2 J ( u n ) , u n ) = lim n ( d n u n 2 β ( 1 q 1 2 ) Ω | x | α | u n | q d x ) d n μ i β C β q / ( 2 q ) d n S n 2 d ( λ 1 ) n 2 d 2 d C β 2 2 q .

However, if β > 0 is given, we can choose λ 1 > 0 so small that for every 0 < λ < λ 1 , the last term on the right-hand side above is greater than 0, which is a contradiction. Similarly, if λ > 0 is given, we can take β 1 > 0 so small that for every 0 < β < β 1 , the last term on the right-hand side above is greater than 0. Then ν i = 0 for each i.

Up to now, we have shown that
lim n Ω | x | 2 b | u n | 2 d x = Ω | x | 2 b | u | 2 d x .
So, by the Breiz-Lieb lemma,
o ( 1 ) u n = u n 2 λ Ω | x | 2 b | u n | 2 d x β Ω | x | α | u n | q d x = u n u 2 u 2 λ Ω | x | 2 b | u | 2 d x β Ω | x | α | u | q d x = u n u 2 + o ( 1 ) u

since J ( u ) = 0 . Thus, we prove that { u n } strongly converges to u in D a 1 , 2 ( Ω ) . □

3 Existence of infinitely many solutions

In this section, we use the minimax procedure to prove the existence of infinitely many solutions. Let Σ be the class of subsets of D a 1 , 2 ( Ω ) { 0 } , which are closed and symmetric with respect to the origin. For A Σ , we define the genus γ ( A ) by
γ ( A ) = min { k N : ϕ C ( A , R k { 0 } ) , ϕ ( x ) = ϕ ( x ) } .
Assume that 1 < q < 2 , then we obtain
J ( u ) = 1 2 Ω | x | 2 a | D u | 2 d x λ 2 Ω | x | 2 b | u | 2 d x λ q Ω | x | α | u | q d x 1 2 u 2 C b λ 2 u 2 β C α q u q .
Define
h ( t ) = 1 2 t 2 λ C 1 t 2 β C 2 t q .
Then, given β > 0 , there exists λ 2 > 0 so small that for every 0 < λ < λ 2 , there exists 0 < T 0 < T 1 such that h ( t ) < 0 for 0 < t < T 0 , h ( t ) > 0 for T 0 < t < T 1 , h ( t ) < 0 for t > T 1 . Similarly, given λ > 0 , we can choose β 2 > 0 with the property that T 0 , T 1 as above exist for each 0 < β < β 2 . Clearly, h ( T 0 ) = h ( T 1 ) = 0 . Following the same idea as in [68], we consider the truncated functional
J ˜ ( u ) = 1 2 Ω | x | 2 a | D u | 2 d x λ 2 ψ ( u ) Ω | x | 2 b | u | 2 d x λ q Ω | x | α | u | q d x ,

where ψ ( u ) = τ ( u ) , and τ : R + [ 0 , 1 ] is a nonincreasing C function such that τ ( t ) = 1 if t T 0 and τ ( t ) = 0 if t T 1 . The main properties of J ˜ are the following.

Lemma 3.1
  1. (1)

    J ˜ C 1 and J ˜ is bounded below.

     
  2. (2)

    If J ˜ ( u ) 0 , then u T 0 and J ˜ ( u ) = J ( u ) .

     
  3. (3)

    For any λ > 0 , there exists β 0 = min { β 1 , β 2 } such that if 0 < β < β 0 and c < 0 , then J ˜ satisfies (PS) c condition.

     
  4. (4)

    for any β > 0 , there exists λ 0 = min { λ 1 , λ 2 } such that if 0 < λ < λ 0 and c < 0 , then J ˜ satisfies (PS) c condition.

     

Proof (1) and (2) are immediate. To prove (3) and (4), observe that all (PS) c sequences for J ˜ with c < 0 must be bounded. Similar to the proof of Lemma 2.3, there exists a convergent subsequence. □

Lemma 3.2 Given m N , there is ε m < 0 such that
γ ( { u D a 1 , 2 ( Ω ) : J ˜ ( u ) ε m } ) m .
Proof Fix m and let H m be an m-dimensional subspace of D a 1 , 2 ( Ω ) . Take u H m , u 0 , write u = r m v with v H m , v = 1 and r m = u . Thus, for 0 < r m < T 0 , since all the norms are equivalent, we have
J ˜ ( u ) = J ( u ) = 1 2 Ω | x | 2 a | D u | 2 d x λ 2 Ω | x | 2 b | u | 2 d x λ q Ω | x | α | u | q d x 1 2 u 2 λ C 1 2 u 2 λ C 2 q u q = 1 2 r m 2 λ C 1 2 r m 2 λ C 2 q r m q : = ε m .
Therefore, we can choose r m ( 0 , T 0 ) so small that J ˜ ( u ) ε m < 0 . Let S r m = { u D a 1 , 2 ( Ω ) : u = r m } , then S r m H m J ˜ ε m . Hence, γ ( J ˜ ε m ) γ ( S r m H m ) = m . Denote Γ m = { A Σ : γ ( A ) m } and let
c m = inf A Γ m sup u A J ˜ ( u ) .

Then < c m ε m < 0 because J ˜ ε m Γ m and J ˜ is bounded from below. □

Lemma 3.3 Let λ, β be as in (3) or (4) of Lemma  3.1. Then all c m are critical values of J ˜ as c m 0 .

Proof It is clear that c m c m + 1 , c m < 0 . Hence, c m c ¯ 0 . Moreover, since all c m are critical values of J ˜ , we claim that c ¯ = 0 . If c ¯ < 0 , because K c ¯ is compact and K c ¯ Σ , it follows that γ ( K c ¯ ) = N 0 < + and there exists δ > 0 such that γ ( K c ¯ ) = γ ( N δ ( K c ¯ ) ) = N 0 . By the deformation lemma there exist ε > 0 ( c ¯ + ε < 0 ) and an odd homeomorphism η such that
η ( J ˜ c ¯ + ε N δ ( K c ¯ ) ) J ˜ c ¯ ε .
Since c m is increasing and converges to c ¯ , there exists m N such that c m > c ¯ ε and c m + N 0 c ¯ and there exists A Γ m + N 0 such that sup u A J ˜ ( u ) < c ¯ + ε . By the properties of γ, we have
γ ( A N δ ( K c ¯ ) ¯ ) γ ( A ) γ ( N δ ( K c ¯ ) ) m , γ ( A N δ ( K c ¯ ) ¯ ) m .
Therefore,
η ( A N δ ( K c ¯ ) ¯ ) Γ m .
Consequently,
sup u η ( A N δ ( K c ¯ ) ¯ ) J ˜ ( u ) c m > c ¯ ε ,

a contradiction, hence c m 0 . □

With Lemma 3.1 to Lemma 3.3, we have proved Theorem 1.1.

Declarations

Acknowledgements

Project is supported by National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11226116), the China Scholarship Council, Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK2012109), the Fundamental Research Funds for the Central Universities (No. JUSRP11118, JUSRP211A22) and Foundation for young teachers of Jiangnan University (No. 2008LQN008).

Authors’ Affiliations

(1)
School of Science, Jiangnan University
(2)
School of Mathematical Science, Nanjing Normal University
(3)
Institute of Science, PLA University of Science and Technology

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