In this paper, we consider a class of quasi-linear elliptic systems of the form

{ − ε p Δ p , A u + V ( x ) | u | p − 2 u = H s ( | u | p , | v | p ) | u | p − 2 u + K ( x ) | u | p ∗ − 2 u , x ∈ R N , − ε p Δ p , A v + V ( x ) | v | p − 2 v = H t ( | u | p , | v | p ) | v | p − 2 v + K ( x ) | v | p ∗ − 2 v , x ∈ R N ,

where Δ p , A u = div ( | ∇ u + i A ( x ) u | p − 2 ( ∇ u + i A ( x ) u ) ) , i is the imaginary unit, A ( x ) : R N → R N is real vector potential, 1 < p < N , V ( x ) is a non-negative potential, p ∗ = N p / ( N − p ) denotes the Sobolev critical exponent for N ≥ 3 and K ( x ) is a bounded positive coefficient.

The scalar case corresponding to (1.1) has received considerable attention in recent years. For p = 2 and A ( x ) ≡ 0 , the scalar case corresponding to (1.1) turns into

− ε 2 Δ u + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , | u | 2 ) u , x ∈ R N .

The equation (1.2) arises in finding standing wave solutions of the nonlinear Schrödinger equation

i ħ ∂ ψ ∂ t = − ħ 2 2 m Δ ψ + W ( x ) ψ − g ( x , | ψ | ) ψ .

A standing wave solution of (1.3) is a solution of the form

ψ ( x , t ) = u ( x ) exp − i E t ħ .

Then ψ ( x , t ) solves (1.3) if and only if u ( x ) solves (1.2) with V ( x ) = W ( x ) − E and ε 2 = ħ 2 2 m .

The equation (1.2) has been extensively investigated in the literature based on various assumptions of the potential V(x) and the nonlinearity f ( x , u ) . See, for example, 123456789101112131415 and the references therein.

There are also many works dealing with the magnetic fields A ( x ) ≠ 0 and p=2 for the scalar case corresponding to (1.1). In 16, the authors firstly obtained the existence of standing waves for special classes of magnetic fields. For many results, we refer the reader to 171819202122.

For general p > 1 , most of the work, as we know, consider the scalar case which corresponds to (1.1) with A(x)≡0. See 2324252627 and the references therein. We especially mention 24 for the existence of positive solutions for a class of p-Laplacian equations. Gloss 24 studied the existence and asymptotic behavior of positive solutions for quasi-linear elliptic equations of the form

− ε p Δ p u + V ( x ) | u | p − 2 u = f ( u ) , x ∈ R N ,

where f is a subcritical nonlinearity without some growth conditions such as the Ambrosetti-Rabinowitz condition. The problem (1.4) has also been studied in 2829303132. The main difficulty in treating this class of equation (1.4) is a possible lack of compactness due to the unboundedness of the domain.

However, to our best knowledge, it seems there is almost no work on the existence of non-trivial solutions to the problem (1.1) involving critical nonlinearity and magnetic fields. We mainly follow the idea of 7. Observe that though the idea was used in other problems, the adaption of the procedure to the problem is not trivial at all. Because of the appearance of magnetic fields A ( x ) , we must deal with the problem for complex-valued functions and therefore we need more delicate estimates.

The outline of the paper is as follows. The forthcoming section is the main result and preliminary results including the appropriate space setting to work with. In Section 3, we study the behavior of ( PS ) c sequence. Section 4 gets that the functional associated to the problem possesses the mountain geometry structure, and the last section concludes the proof of the main result.

Firstly, we make the following assumptions on V(x), A(x), H ( s , t ) and K(x) throughout the paper:

( V 0 ) V ∈ C ( R N , R ) , V ( 0 ) = inf x ∈ R N V ( x ) = 0 and there exists b > 0 such that the set ν b : = { x ∈ R N : V ( x ) < b } has finite Lebesgue measure;

( A 0 ) A ∈ C ( R N , R N ) and A ( 0 ) = 0 ;

( K 0 ) K ( x ) ∈ C ( R N , R + ) , 0 < inf K ≤ sup K < ∞ ;

( H 1 ) H ( s , t ) ∈ C 1 ( R + × R + , R ) and H s , H t = o ( | s | + | t | ) as | s | + | t | → 0 ;

( H 2 ) there exist c 1 > 0 and p < α < p ∗ such that

| H s ( s , t ) | , | H t ( s , t ) | ≤ c 1 ( 1 + | s | α − p p + | t | α − p p ) ;

( H 3 ) there are a 0 > 0 , θ ∈ ( p , p ∗ ) and α , β > p such that H ( s , t ) ≥ p a 0 ( | s | α p + | t | β p ) and 0 < θ p H ( s , t ) ≤ s H s + t H t .

Under the above mentioned conditions, we get the following result.

Theorem 1 Suppose that the assumptions (V0), (A0), (K0) and (H1)-(H3) hold. Then for any σ > 0 , there is ε σ > 0 such that if ε < ε σ , the problem (1.1) has at least one solution ( u ε , v ε ) which satisfies

θ − p p θ ∫ R N ( ε p ( | ∇ | u ε | | p + | ∇ | v ε | | p ) + V ( x ) | u ε | p + V ( x ) | v ε | p ) ≤ σ ε N .

Setting λ = ε − p , the problem (1.1) is equivalent to the following problem:

{ − Δ p , A u + λ V ( x ) | u | p − 2 u = λ H s ( | u | p , | v | p ) | u | p − 2 u + λ K ( x ) | u | p ∗ − 2 u , x ∈ R N , − Δ p , A v + λ V ( x ) | v | p − 2 v = λ H t ( | u | p , | v | p ) | v | p − 2 v + λ K ( x ) | v | p ∗ − 2 v , x ∈ R N .

We are going to prove the following result.

Theorem 2 Suppose that the assumptions (V0), (A0), (K0) and (H1)-(H3) hold. Then for any σ>0, there is Λ σ > 0 such that if λ > Λ σ , the problem (2.1) has at least one solution ( u λ , v λ ) which satisfies

θ − p p θ ∫ R N ( ( | ∇ | u λ | | p + | ∇ | v λ | | p ) + λ V ( x ) | u λ | p + λ V ( x ) | v λ | p ) ≤ σ λ 1 − p N .

For convenience, we quote the following notations. Let E λ , A denote the Banach space

E λ , A = { u ∈ W 1 , p ( R N ) : ∫ R N λ V ( x ) | u | p < ∞ } , λ > 0

equipped with the norm

∥ u ∥ λ , A = ( ∫ R N ( | ∇ u + i λ 1 p A ( x ) u | p + λ V ( x ) | u | p ) ) 1 p .

Set E = E λ , A × E λ , A and ∥ ( u , v ) ∥ E p = ∥ u ∥ λ , A p + ∥ v ∥ λ , A p for any ( u , v ) ∈ E .

Similar to the diamagnetic inequality 16, we have

| ∇ | u ( x ) | | ≤ | ∇ u u ¯ | u | | = | Re ( ∇ u + i λ 1 p A u ) u ¯ | u | | ≤ | ∇ u + i λ 1 p A u |

(the bar denotes complex conjugation). This inequality shows that if u ∈ E λ , A , then | u | ∈ W 1 , p ( R N ) and therefore u ∈ L q ( R N ) for any q ∈ [ p , p ∗ ) . That is to say, if u n ⇀ u in Eλ,A, then u n → u in L loc q ( R N ) for any q∈[p,p∗) and un→u a.e. in R N .

The energy functional associated with (2.1) is defined by

I λ ( u , v ) = 1 p ∫ R N ( | ∇ u + i λ 1 p A u | p + λ V ( x ) | u | p + | ∇ v + i λ 1 p A v | p + λ V ( x ) | v | p ) − λ p ∗ ∫ R N K ( x ) ( | u | p ∗ + | v | p ∗ ) − λ p ∫ R N H ( | u | p , | v | p ) = 1 p ∥ ( u , v ) ∥ E p − λ ∫ R N G ( u , v ) ,

where G ( u , v ) = 1 p ∗ K ( x ) ( | u | p ∗ + | v | p ∗ ) + 1 p H ( | u | p , | v | p ) .

Under the assumptions of Theorem 2, standard arguments 33 show that I λ ∈ C 1 ( E λ , A , R ) and its critical points are weak solutions of the equation (2.1).

We call a sequence { ( u n , v n ) } ⊂ E a (PS)c sequence if I λ ( u n , v n ) → c and I λ ′ ( u n , v n ) → 0 strongly in E ′ (E′ is the dual space of E). I λ is said to satisfy the (PS)c condition if any (PS)c sequence contains a convergent subsequence.

The main result of Section 3 is the following compactness result.

Proposition 3.1 Let the assumptions of Theorem 2 be satisfied. There exists a constant α 0 ≥ 0 independent of λ such that, for any (PS)c sequence {(un,vn)}⊂E for Iλ with ( u n , v n ) ⇀ ( u , v ) , either ( u n , v n ) → ( u , v ) or c − I λ ( u , v ) ≥ α 0 λ 1 − N p .

As a consequence, we obtain the following result.

Proposition 3.2 Assume that the assumptions of Proposition 3.1 hold, I λ ( u , v ) satisfies the (PS)c condition for all c ≤ α 0 λ 1 − N p .

In order to prove Proposition 3.1, we need the following lemmas.

Lemma 3.1 Let the assumptions of Theorem 2 be satisfied. { ( u n , v n ) } is a (PS)c sequence of Iλ. Then c ≥ 0 and {(un,vn)} is bounded in the space E.

Proof

One has

I λ ( u n , v n ) − 1 θ I λ ′ ( u n , v n ) ( u n , v n ) = ( 1 p − 1 θ ) ∥ ( u n , v n ) ∥ E p + ( 1 θ − 1 p ∗ ) λ ∫ R N K ( x ) ( | u n | p ∗ + | v n | p ∗ ) + λ ∫ R N ( 1 θ ( | u n | p H s ( | u n | p , | v n | p ) + | v n | p H t ( | u n | p , | v n | p ) ) − 1 p H ( | u n | p , | v n | p ) ) ≥ ( 1 p − 1 θ ) ∥ ( u n , v n ) ∥ E p .

Together with Iλ(un,vn)→c and I λ ′ ( u n , v n ) → 0 as n → ∞ , we have

( 1 p − 1 θ ) ∥ ( u n , v n ) ∥ E p ≤ c + o ( 1 ) + ε n ∥ ( u n , v n ) ∥ E .

Then {(un,vn)} is bounded and c≥0. □

From Lemma 3.1, we may assume (un,vn)⇀(u,v) in E and (un,vn)→(u,v) in L loc q ( R N ) × L loc q ( R N ) for any q∈[p,p∗) and un→u, v n → v a.e. in RN.

Lemma 3.2 Let γ ∈ [ p , p ∗ ) . There is a subsequence { ( u n j , v n j ) } such that for any ε > 0 , there is r ε > 0 with r ≥ r ϵ

lim j → ∞ sup ∫ B j ∖ B r ( | u n j | γ + | v n j | γ ) ≤ ε ,

where B r : = { x ∈ R N : | x | ≤ r } .

Proof The proof of Lemma 3.2 is similar to that of Lemma 3.2 of 27, so we omit it. □

Let η ∈ C ∞ ( R + ) be a smooth function satisfying 0 ≤ η ( t ) ≤ 1 , η ( t ) = 1 if t ≤ 1 and η ( t ) = 0 if t ≥ 2 . Define u ˜ j ( x ) = η ( 2 | x | / j ) u ( x ) , v ˜ j ( x ) = η ( 2 | x | / j ) v ( x ) . Obviously, we have

∥ u − u ˜ j ∥ E λ , A → 0 and ∥ v − v ˜ j ∥ E λ , A → 0 as  j → ∞ .

Lemma 3.3 One has

lim j → ∞ Re ∫ R N ( H s ( | u n j | p , | v n j | p ) | u n j | p − 2 u n j − H s ( | u n j − u ˜ j | p , | v n j − v ˜ j | p ) | u n j − u ˜ j | p − 2 ( u n j − u ˜ j ) − H s ( | u ˜ j | p , | v ˜ j | p ) | u ˜ j | p − 2 u ˜ j ) φ ¯ = 0

and

lim j → ∞ Re ∫ R N ( H t ( | u n j | p , | v n j | p ) | v n j | p − 2 v n j − H t ( | u n j − u ˜ j | p , | v n j − v ˜ j | p ) | v n j − v ˜ j | p − 2 ( v n j − v ˜ j ) − H t ( | u ˜ j | p , | v ˜ j | p ) | v ˜ j | p − 2 v ˜ j ) ψ ¯ = 0

uniformly in ( φ , ψ ) ∈ E with ∥ ( φ , ψ ) ∥ E ≤ 1 .

Proof The local compactness of Sobolev embedding implies that for any r ≥ 0 , we have

lim j → ∞ Re ∫ B r ( H s ( | u n j | p , | v n j | p ) | u n j | p − 2 u n j − H s ( | u n j − u ˜ j | p , | v n j − v ˜ j | p ) | u n j − u ˜ j | p − 2 ( u n j − u ˜ j ) − H s ( | u ˜ j | p , | v ˜ j | p ) | u ˜ j | p − 2 u ˜ j ) φ ¯ = 0

uniformly in ∥ φ ∥ E λ , A ≤ 1 . For any ε > 0 , there exists rε>0 such that

lim j → ∞ sup ∫ B j ∖ B r | u ˜ j | γ ≤ ∫ R N | u | γ ≤ ε

for all r≥rϵ. Together with the assumption (H2) and the Hölder inequality, it follows from Lemma 3.2 that

lim j → ∞ sup Re ∫ R N ( H s ( | u n j | p , | v n j | p ) | u n j | p − 2 u n j − H s ( | u n j − u ˜ j | p , | v n j − v ˜ j | p ) | u n j − u ˜ j | p − 2 ( u n j − u ˜ j ) − H s ( | u ˜ j | p , | v ˜ j | p ) | u ˜ j | p − 2 u ˜ j ) φ ¯ = lim j → ∞ sup Re ∫ B j ∖ B r ( H s ( | u n j | p , | v n j | p ) | u n j | p − 2 u n j − H s ( | u n j − u ˜ j | p , | v n j − v ˜ j | p ) | u n j − u ˜ j | p − 2 ( u n j − u ˜ j ) − H s ( | u ˜ j | p , | v ˜ j | p ) | u ˜ j | p − 2 u ˜ j ) φ ¯ ≤ c 1 lim j → ∞ sup ∫ B j ∖ B r ( | u n j | p − 1 + | v n j | p − 1 + | u ˜ j | p − 1 + | v ˜ j | p − 1 ) | φ ¯ | + c 2 lim j → ∞ sup ∫ B j ∖ B r ( | u n j | α − 1 + | v n j | α − 1 + | u ˜ j | α − 1 + | v ˜ j | α − 1 ) | φ ¯ | ≤ c 1 lim j → ∞ sup ( ∥ u n j ∥ L p ( B j ∖ B r ) p − 1 + ∥ v n j ∥ L p ( B j ∖ B r ) p − 1 + ∥ u ˜ j ∥ L p ( B j ∖ B r ) p − 1 + ∥ v ˜ j ∥ L p ( B j ∖ B r ) p − 1 ) ∥ φ ¯ ∥ L p ( B j ∖ B r ) + c 2 lim j → ∞ sup ( ∥ u n j ∥ L α ( B j ∖ B r ) α − 1 + ∥ v n j ∥ L α ( B j ∖ B r ) α − 1 + ∥ u ˜ j ∥ L α ( B j ∖ B r ) α − 1 + ∥ v ˜ j ∥ L α ( B j ∖ B r ) α − 1 ) ∥ φ ¯ ∥ L α ( B j ∖ B r ) ≤ c 3 ε p − 1 p + c 4 ε α − 1 α ,

where c i (i = 1 , 2 , 3 , 4 ) are positive constants. Similarly, we can prove

lim j → ∞ Re ∫ R N ( H t ( | u n j | p , | v n j | p ) | v n j | p − 2 v n j − H t ( | u n j − u ˜ j | p , | v n j − v ˜ j | p ) | v n j − v ˜ j | p − 2 ( v n j − v ˜ j ) − H t ( | u ˜ j | p , | v ˜ j | p ) | v ˜ j | p − 2 v ˜ j ) ψ ¯ = 0 .

 □

Lemma 3.4 Let {(un,vn)} and { ( u ˜ n , v ˜ n ) } be as defined above. Then the following conclusions hold:

I λ ( u n − u ˜ n , v n − v ˜ n ) → c − I λ ( u , v )

and

I λ ′ ( u n − u ˜ n , v n − v ˜ n ) → 0 in  E ′ ( the dual space of  E ) .

Proof By using the similar arguments of 3435, we have

I λ ( u n − u ˜ n , v n − v ˜ n ) = I λ ( u n , v n ) − I λ ( u ˜ n , v ˜ n ) + λ p ∗ ∫ R N K ( x ) ( ( | u n | p ∗ − | u n − u ˜ n | p ∗ − | u ˜ n | p ∗ ) + ( | v n | p ∗ − | v n − v ˜ n | p ∗ − | v ˜ n | p ∗ ) ) + λ p ∫ R N ( H ( | u n | p , | v n | p ) − H ( | u n − u ˜ n | p , | v n − v ˜ n | p ) − H ( | u ˜ n | p , | v ˜ n | p ) ) + o ( 1 ) .

By (3.1) and the similar idea of proving the Brézis-Lieb lemma 36, it is easy to get

lim n → ∞ ∫ R N K ( x ) ( ( | u n | p ∗ − | u n − u ˜ n | p ∗ − | u ˜ n | p ∗ ) + ( | v n | p ∗ − | v n − v ˜ n | p ∗ − | v ˜ n | p ∗ ) ) = 0

and

lim n → ∞ ∫ R N ( H ( | u n | p , | v n | p ) − H ( | u n − u ˜ n | p , | v n − v ˜ n | p ) − H ( | u ˜ n | p , | v ˜ n | p ) ) = 0 .

Furthermore, using the fact Iλ(un,vn)→c and I λ ( u ˜ n , v ˜ n ) → I λ ( u , v ) , we obtain

I λ ( u n − u ˜ n , v n − v ˜ n ) → c − I λ ( u , v ) .

In order to prove I λ ′ ( u n − u ˜ n , v n − v ˜ n ) → 0 in E − 1 , for any (φ,ψ)∈E, it follows that

I λ ′ ( u n − u ˜ n , v n − v ˜ n ) ( φ , ψ ) = I λ ′ ( u n , v n ) ( φ , ψ ) − I λ ′ ( u ˜ n , v ˜ n ) ( φ , ψ ) + λ Re ∫ R N K ( x ) ( | u n | p ∗ − 2 u n − | u n − u ˜ n | p ∗ − 2 ( u n − u ˜ n ) − | u ˜ n | p ∗ − 2 u ˜ n ) φ ¯ + λ Re ∫ R N K ( x ) ( | v n | p ∗ − 2 v n − | v n − v ˜ n | p ∗ − 2 ( v n − v ˜ n ) − | v ˜ n | p ∗ − 2 v ˜ n ) ψ ¯ + λ Re ∫ R N ( H s ( | u n | p , | v n | p ) | u n | p − 2 u n − H s ( | u n − u ˜ n | p , | v n − v ˜ n | p ) | u n − u ˜ n | p − 2 ( u n − u ˜ n ) − H s ( | u ˜ n | p , | v ˜ n | p ) | u ˜ n | p − 2 u ˜ n ) φ ¯ + λ Re ∫ R N ( H t ( | u n | p , | v n | p ) | v n | p − 2 v n − H t ( | u n − u ˜ n | p , | v n − v ˜ n | p ) | v n − v ˜ n | p − 2 ( v n − v ˜ n ) − H t ( | u ˜ n | p , | v ˜ n | p ) | v ˜ n | p − 2 v ˜ n ) ψ ¯ + o ( 1 ) .

It is standard to check that

lim n → ∞ ∫ R N K ( x ) ( | u n | p ∗ − 2 u n − | u n − u ˜ n | p ∗ − 2 ( u n − u ˜ n ) − | u ˜ n | p ∗ − 2 u ˜ n ) φ ¯ = 0

and

lim n → ∞ ∫ R N K ( x ) ( | v n | p ∗ − 2 v n − | v n − v ˜ n | p ∗ − 2 ( v n − v ˜ n ) − | v ˜ n | p ∗ − 2 v ˜ n ) ψ ¯ = 0

uniformly in (φ,ψ)∈E with ∥(φ,ψ)∥E≤1. Together with Lemma 3.3, we have

I λ ′ ( u n − u ˜ n , v n − v ˜ n ) → 0 in  E ′ .

 □

Let u n 1 = u n − u ˜ n , v n 1 = v n − v ˜ n , then u n − u = u n 1 + ( u ˜ n − u ) , v n − v = v n 1 + ( v ˜ n − v ) . From (3.1), we get (un,vn)→(u,v) in E if and only if ( u n 1 , v n 1 ) → ( 0 , 0 ) in E.

Observe that

I λ ( u n 1 , v n 1 ) − 1 p I λ ′ ( u n 1 , v n 1 ) ( u n 1 , v n 1 ) = ( 1 p − 1 p ∗ ) λ ∫ R N K ( x ) ( | u n 1 | p ∗ + | v n 1 | p ∗ ) + λ p ∫ R N ( | u n 1 | p H s ( | u n 1 | p , | v n 1 | p ) + | v n 1 | p H t ( | u n 1 | p , | v n 1 | p ) − H ( | u n 1 | p , | v n 1 | p ) ) ≥ λ N ∫ R N K ( x ) ( | u n 1 | p ∗ + | v n 1 | p ∗ ) ≥ λ N K min ∫ R N ( | u n 1 | p ∗ + | v n 1 | p ∗ ) ,

where K min = inf x ∈ R N K ( x ) > 0 . Furthermore, we get

∥ ( u n 1 , v n 1 ) ∥ p ∗ p ∗ ≤ N ( c − I λ ( u , v ) ) λ K min + o ( 1 ) .

Now, we consider the energy level of the functional Iλ below which the (PS)c condition holds.

Let V b ( x ) : = max { V ( x ) , b } , where b is a positive constant in the assumption (V0). Since the set ν b has finite measure, we get

∫ R N V ( x ) ( | u n 1 | p + | v n 1 | p ) = ∫ R N V b ( x ) ( | u n 1 | p + | v n 1 | p ) + o ( 1 ) .

In connection with the assumptions (H1)-(H3) and the Young inequality, there exists C b > 0 such that

Let S be the best Sobolev constant of the immersion

S ∥ u ∥ p ∗ p ≤ ∫ R N | ∇ u | p for all  u ∈ W 1 , p ( R N ) .

Proof of Proposition 3.1 Assume that ( u n , v n ) ↛ ( u , v ) , then

lim inf n → ∞ ∥ ( u n 1 , v n 1 ) ∥ E > 0

and

c − I λ ( u , v ) > 0 .

By the Sobolev embedding inequality and the diamagnetic inequality, we get

S ( ∥ u n 1 ∥ p ∗ p + ∥ v n 1 ∥ p ∗ p ) ≤ ∫ R N ( | ∇ u n 1 | p + | ∇ v n 1 | p ) ≤ ∫ R N ( | ∇ u n 1 + i λ 1 p A ( x ) u n 1 | p + | ∇ v n 1 + i λ 1 p A ( x ) v n 1 | p ) = ∫ R N ( | ∇ u n 1 + i λ 1 p A ( x ) u n 1 | p + λ V ( x ) | u n 1 | p + | ∇ v n 1 + i λ 1 p A ( x ) v n 1 | p + λ V ( x ) | v n 1 | p ) − λ ∫ R N V ( x ) ( | u n 1 | p + | v n 1 | p ) = λ ∫ R N K ( x ) ( | u n 1 | p ∗ + | v n 1 | p ∗ ) + | u n 1 | p H s ( | u n 1 | p , | v n 1 | p ) + | v n 1 | p H t ( | u n 1 | p , | v n 1 | p ) − λ ∫ R N V b ( x ) ( | u n 1 | p + | v n 1 | p ) + o ( 1 ) ≤ λ b ( ∥ u n 1 ∥ p p + ∥ v n 1 ∥ p p ) + λ C b ( ∥ u n 1 ∥ p ∗ p ∗ + ∥ v n 1 ∥ p ∗ p ∗ ) − λ b ( ∥ u n 1 ∥ p p + ∥ v n 1 ∥ p p ) + o ( 1 ) = λ C b ( ∥ u n 1 ∥ p ∗ p ∗ + ∥ v n 1 ∥ p ∗ p ∗ ) + o ( 1 ) .

This, together with (3.2), gives

S ≤ λ C b ( ∥ u n 1 ∥ p ∗ p ∗ + ∥ v n 1 ∥ p ∗ p ∗ ) p ∗ − p p ∗ + o ( 1 ) ≤ λ C b ( N ( c − I λ ( u , v ) ) λ K min ) p N + o ( 1 ) = λ 1 − p N C b ( N K min ) p N ( c − I λ ( u , v ) ) p N + o ( 1 ) .