Bull. Malays. Math. Sci. Soc. DOI 10.1007/s40840-015-0253-7

Quasilinear Elliptic Systems Involving Critical Hardy–Sobolev and Sobolev Exponents Dongsheng Kang1 · Yangguang Kang2

Received: 11 January 2014 / Revised: 28 March 2014 © Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015

Abstract In this paper, a system of quasilinear elliptic equations is investigated, which involves critical exponents and multiple Hardy-type terms. By variational methods and analytic techniques, the existence of positive solutions to the system is established. The conclusions are new even when the Hardy-type terms disappear. Keywords Quasilinear elliptic system · Positive solution · Critical exponent · Hardy inequality · Variational method Mathematics Subject Classification

35J47 · 35J62

Communicated by Norhashidah M. Ali.

B

Dongsheng Kang [email protected] Yangguang Kang [email protected]

1

School of Mathematics and Statistics, South–Central University For Nationalities, Wuhan 430074, People’s Republic of China

2

School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China

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D. Kang, Y. Kang

1 Introduction In this paper, we study the following elliptic system: ⎧ u p−1 η1 ⎪ ⎪ − p u − μ = ∗ Hu (u, v) + ⎪ ⎪ ⎨ |x| p p v p−1 η1 ⎪ = ∗ Hv (u, v) + − p v − μ ⎪ ⎪ |x| p p ⎪ ⎩ u, v > 0, (u, v) ∈ D × D,

η2 Q u (u, v) , p ∗ (t) |x|t η2 Q v (u, v) , p ∗ (t) |x|t

(1.1)

 p where 1 < p < N , 0 ≤ μ < μ¯ := (N − p)/ p , 0 < t < p, η1 > 0, η2 > 0, − p · := −div(|∇ · | p−2 ·) is the p–Laplace operator, D 1, p (R N )  the spacep D := ∞ N 1/ p denotes the completion of C0 (R ) with respect to ( R N |∇ · | dx) , μ¯ is the best Hardy constant, p ∗ := N p/(N − p) is the critical Sobolev exponent, and p ∗ (t) := p(N − t)/(N − p) is the critical Hardy–Sobolev exponent with p ∗ (0) = p ∗ . Hu , Hv , Q u , and Q v are the partial derivatives of the 2–variable C 1 –functions H (u, v) and Q(u, v), respectively. The functions H and Q satisfy the following conditions: (H) H, Q ∈ C 1 (R+ × R+ , R+ ), Hu (u, 0) = Hu (0, v) = Hv (u, 0) = Hv (0, v) = 0, ∀ u, v ≥ 0, Q u (u, 0) = Q u (0, v) = Q v (u, 0) = Q v (0, v) = 0, ∀ u, v ≥ 0, ∗ H (λu, λv) = λ p H (u, v), ∀ λ ≥ 0, u, v ≥ 0, ( p ∗ –homogeneity), Q(λu, λv) = λ p

∗ (t)

Q(u, v), ∀ λ ≥ 0, u, v ≥ 0, ( p ∗ (t)–homogeneity),

and the 1–homogenous functions G and G¯ are concave, where G and G¯ are defined as follows: ∗ ∗ ¯ p∗ (t) , β p∗ (t) ) = Q(α, β), ∀ α, β ≥ 0. G(α p , β p ) = H (α, β), G(α

The following properties are important and well known: (H ) Suppose F(s, t) is a q-homogeneous differential function with q ≥ 1. Then (i) s Fs (s, t) + t Ft (s, t) = q F(s, t), ∀s, t ∈ R; (ii) C F is attained at some (s0 , t0 ) ∈ R2 , where C F := max{F(s, t)|s, t ∈ R, |s|q + |t|q = 1}; (iii) |F(s, t)| ≤ C F (|s|q + |t|q ), ∀ s, t ∈ R; (iv) Fs (s, t) and Ft (s, t) are (q − 1)-homogeneous. In this paper, we work in the product space D × D. The corresponding energy functional of (1.1) is defined on D × D by

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On the Quasilinear Elliptic Systems Involving Critical

I (u, v) :=



1 |u| p + |v| p  |∇u| p + |∇v| p − μ dx p R N |x| p η1 Q(|u|, |v|) η2 − ∗ H (|u|, |v|)dx − ∗ dx. N N p R p (t) R |x|t

Then I ∈ C 1 (D × D, R). A pair of functions (u, v) ∈ D × D is said to be a solution of (1.1) if u, v > 0, and (u, v) = (0, 0),

I (u, v), (ϕ, φ) = 0, ∀ (ϕ, φ) ∈ D × D,

where I (u, v) denotes the Fréchet derivative of I at (u, v). Problem (1.1) is related to the Hardy and Hardy–Sobolev inequalities [8,20]):

|u| p 1 dx ≤ |∇u| p dx , ∀ u ∈ C0∞ (R N ) , (1.2) p N N |x| μ ¯ R R

|u| p∗ (t)  ∗p p (t) dx ≤ C( p, t) |∇u| p dx , ∀ u ∈ C0∞ (R N ) , (1.3) t R N |x| RN

where C( p, t) is a constant depending on p and t, 1 < p < N and 0 ≤ t < p. By (1.2) the operator L := (− p · −μ| · | p−2 ·/|x| p ) is positive for all μ < μ, ¯ and therefore, the following equivalent norm of D can be defined:



u :=

|u| p   1p |∇u| p − μ dx , ∀ u ∈ D. |x| p RN

Suppose (H) holds. By (H ), (1.2) and (1.1), the following best Hardy–Sobolev constants are well defined:

|u| p  |∇u| p − μ dx |x| p RN S(μ, t) := inf , (1.4) ∗ p u∈D \{0}

|u| p (t)  p∗ (t) dx t R N |x|

|u| p + |v| p  |∇u| p + |∇v| p − μ dx |x| p RN S H (μ, 0) := inf , (1.5)

 p∗ u,v∈D \{0} p H (|u|, |v|)dx RN

|u| p + |v| p  |∇u| p + |∇v| p − μ dx |x| p RN , (1.6) S Q (μ, t) := inf p

Q(|u|, |v|)  p∗ (t) u,v∈D \{0} dx |x|t RN where 0 ≤ t < p, −∞ < μ < μ. ¯ It should be mentioned that the strongly coupled dx are critical in the senses of Sobolev or terms R N H (|u|, |v|)dx and R N Q(|u|,|v|) |x|t Hardy–Sobolev embedding. Morais Filho et al. studied the constant S H (0, 0) and

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proved the existence of solutions for a quasilinear elliptic systems in [17]. Alves et al. studied in [3] the following best constant and found its extremals:   |∇u|2 + |∇v|2 dx RN A(σ, τ ) := inf (1.7)

 2∗ , u,v∈D 1,2 (R N )\{0} 2 σ τ |u| |v| dx RN

where 1 < σ, τ < 2∗ − 1, σ + τ = 2∗ := 2N /(N − 2). Note that A(σ, τ ) in (1.7) is a special case of S H (0, 0). The methods and conclusions in [3] and [17] are very stimulating. In recent years, much attention has been paid to the semilinear and quasilinear elliptic problems involving the Hardy and Hardy–Sobolev inequalities, and many results were obtained providing us very good insight into the problems (e.g., [1,5,6,9– 11,14,15,18,19,22,23,30,32,33], and the references therein). In particular, Filippucci et al. studied in [18] the following problem: ⎧ ∗ u p−1 u p (s)−1 ⎪ p ∗ −1 ⎪ ⎨ − p u − μ =u + , |x| p |x|s N ⎪ ⎪ u ∈ D, u > 0 in R , ⎩ −∞ < μ < μ, ¯ 0 < s < p.

(1.8)

The main difficulty of studying (1.8) is that the critical Hardy–Sobolev and Sobolev exponents appear simultaneously in the equation and induce more difficulties. By very technic and complicated analysis, the authors of [18] proved the existence of positive solutions to (1.8) by the Mountain–Pass theorem [4] and the concentration compactness principle [26,27]. The extremals of the best constant S(μ, t) in (1.4) and some related singular quasilinear elliptic problems were investigated in [1,18,19] and [23], and we infer that, for all 0 ≤ t < p, 0 ≤ μ < μ, ¯ the best constant Sμ,t is achieved by the implicit extremal function: ε (x) = ε Vμ,t

p−N p

Uμ,t (ε−1 x) , ∀ ε > 0 ,

(1.9)

which satisfies

ε (x)| p  ε (x)| p ∗ (t)

|Vμ,t |Vμ,t N −t ε p |∇Vμ,t (x)| − μ = = (Sμ,t ) p−t , p t N N |x| |x| R R

where Uμ,t (x) is some radial function. On the other hand, the singular elliptic systems involving the Hardy and Hardy– Sobolev inequalities have been seldom studied, we can only find several results in [2,7,16,21,24,25,28] and [29], where some nonlinear singular critical systems were investigated, the corresponding best Hardy–Sobolev constants were studied and existence results of solutions were obtained. The main difficulties of studying singular elliptic systems are that the singularity may occur and the strongly coupled terms may cause more difficulties.

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On the Quasilinear Elliptic Systems Involving Critical

To continue, we define p  M H := max H (|α|, |β|) p∗ α, β ∈ R, |α| p + |β| p = 1 ; p  M Q := max Q(|α|, |β|) p∗ (t) α, β ∈ R, |α| p + |β| p = 1 .

(1.10) (1.11)

Then there exist (αi , βi ) ∈ R+ × R+ , i = 1, 2, such that M H and M Q are achieved respectively, that is, p

p

p

M H = H (α1 , β1 ) p∗ , α1 + β1 = 1,

(1.12)

p p∗ (t)

(1.13)

M Q = Q(α2 , β2 )

p

p

, α2 + β2 = 1.

In this paper, stimulated by the references mentioned above, we investigate (1.1). The main results of this paper are summarized in the following theorems. To the best of our knowledge, the conclusions are new even in the case μ = 0. Theorem 1.1 Suppose that 0 ≤ t < p, −∞ < μ < μ¯ and (H) holds. Then −1 −1 S(μ, 0), S Q (μ, t) = M Q S(μ, t). (i) S H (μ, 0) = M H   ε (x), β V ε (x) , (ii) For all 0 ≤ μ < μ, ¯ S H (μ, 0) has the minimizers α1 Vμ,0 1 μ,0   ε (x), β V ε (x) , where V ε (x) are defined S Q (μ, t) has the minimizers α2 Vμ,t 2 μ,t μ,t as in (1.9).

Theorem 1.2 Suppose that 1 < p < N , 0 ≤ μ < μ, ¯ 0 < t < p, η1 > 0, η2 > 0 and (H) holds. Then the problem (1.1) has a solution. Remark 1.1 The coefficients 1/ p ∗ and 1/ p ∗ (t) in (1.1) are only used for the convenience of computation and have no particular meanings. By Theorem 1.1, the existence of solutions to (1.1) is obvious in anyone of the following cases: (i) η1 = 0, η2 > 0, t ≥ 0; (ii) η1 > 0, η2 = 0, t ≥ 0; (iii) t = 0, η1 > 0, η2 > 0. Remark 1.2 The following problem is an example of (1.1) : ⎧ u p−1 σ1 ⎪ ⎪ = ∗ u σ1 −1 v τ1 + ⎪ − p u − μ ⎪ ⎨ |x| p p v p−1 τ1 σ1 τ1 −1 ⎪ = ∗u v + − p v − μ ⎪ ⎪ |x| p p ⎪ ⎩ u, v > 0, (u, v) ∈ D × D,

σ2 u σ2 −1 v τ2 , p ∗ (t) |x|t σ τ −1 2 2 τ2 u v , ∗ p (t) |x|t

(1.14)

where the parameters satisfy the following condition: 



(H ) N ≥ 3, 1 < p < N , 0 ≤ μ < μ¯ = i = 1, 2, σ1 + τ1 = p ∗ =

N−p p

p , 0 < t < p, σi , τi > 1,

Np p(N − t) , σ2 + τ2 = p ∗ (t) = . N−p N−p

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Note that (1.14) involves the critical Hardy–Sobolev and Sobolev exponents and admits a solution by Theorem 1.2. This paper is organized as follows: Theorem 1.1 is verified in Sect. 2, and some preliminary results are established  in Sect. 3, and Theorem 1.2 is proved in Sect. 4. In the following argument, u = ( R N (|∇u| p − μ|u| p |x|− p )dx)1/ p denotes the equivalent norm of the space D, and (u, v) D×D = ( u p + v p )1/ p is the norm of the space D × D. For all ε > 0 small enough, O(εt ) denotes the quantity satisfying |O(εt )|/εt ≤ C, o(εt ) means |o(εt )|/εt → 0 as ε → 0 and o(1) is a generic infinitesimal value. In particular, the quantity O1 (εt ) means that there exist the constants C1 , C2 > 0 such that C1 εt ≤ O1 (εt ) ≤ C2 εt as ε small. We always denote positive constants as C and omit dx in integrals for convenience.

2 The Best Constants SH (µ, 0) and S Q (µ, t) In this section, we study S H (μ, 0) and S Q (μ, t) and verify Theorem 1.1. Proof of Theorem 1.1 (i) We only show the proof for S Q (μ, t). The argument is similar to that of [17], where the best constant S H (0, 0) was studied. Let w ∈ D\{0} and (α2 , β2 ) be defined as in (1.13). Choosing (u, v) = (α2 w, β2 w) in (1.6) we have

|w| p  |∇w| p − μ p |x| RN ≥ S Q (μ, t). p ∗ (t)  ∗p p

|w| p (t) ∗ (t) p |Q(α2 , β2 )| |x|t RN

(|α2

|p

+ |β2

|p)

(2.1)

Taking the infimum as w ∈ D \ {0} in (2.1), by (1.4) and (1.10)–(1.13) we have −1 MQ S(μ, t) ≥ S Q (μ, t).

(2.2)

For any u, v ∈ D \ {0}, by Proposition 1 of [17] we have that

 − t  Q(|u|, |v|) − t = Q |x| p∗ (t) |u|, |x| p∗ (t) |v| t |x| RN RN   − t − t ≤ Q |x| p∗ (t) u L p∗ (t) (R N ) , |x| p∗ (t) v L p∗ (t) (R N ) .

(2.3)

Set − 1

p − t − t p p . θ := |x| p∗ (t) u L p∗ (t) (R N ) + |x| p∗ (t) v L p∗ (t) (R N ) Then

θ |x|

123

− p∗t(t)

p

u L p∗ (t) (R N ) + θ |x|

− p∗t(t)

p

v L p∗ (t) (R N ) = 1.

(2.4)

On the Quasilinear Elliptic Systems Involving Critical

From (1.11), (1.13), (2.3), and (2.4) it follows that

|u| p + |v| p  |∇u| p + |∇v| p − μ |x| p RN p

 Q(|u|, |v|) α+β

RN

≥ S(μ, t)

|x| t

∗ ∗ p p |u| p (t)  p∗ (t)

|v| p (t)  p∗ (t) + t t R N |x| R N |x|

  p∗p(t) − p∗t(t) − p∗t(t) ∗ ∗ Q |x| u L p (t) (R N ) , |x| v L p (t) (R N )

|x|

− p∗t(t)

p

− p∗t(t)

p

v L p∗ (t) (R N ) = S(μ, t)

  p∗p(t) − t − t Q |x| p∗ (t) u L p∗ (t) (R N ) , |x| p∗ (t) v L p∗ (t) (R N )

θ |x|

u L p∗ (t) (R N ) + |x|

− p∗t(t)

p

u L p∗ (t) (R N ) + θ |x|

− p∗t(t)

p

v L p∗ (t) (R N ) = S(μ, t)

  p∗p(t) − t − t Q θ |x| p∗ (t) u L p∗ (t) (R N ) , θ |x| p∗ (t) v L p∗ (t) (R N ) 1 −1 ≥ p S(μ, t) = M Q S(μ, t). ∗ (t) p |Q(α2 , β2 )| Taking the infimum as u, v ∈ D \ {0} we have −1 MQ S(μ, t) ≤ S Q (μ, t),

which together with (2.2) implies that −1 S Q (μ, t) = M Q S(μ, t).

 

(ii) From (i), (1.5), and (1.6) the desired result follows.

3 Appropriate Palais–Smale Sequence To find positive solutions of (1.1), we define the functional J on D × D by J (u, v) :=

1 Q(u + , v+ ) η1 η2

(u, v) p − ∗ H (u + , v+ ) − ∗ , p p RN p (t) R N |x|t

where w+ = max{w, 0} for all w ∈ D. Then J ∈ C 1 (D × D, R) according to (H) and a solution of (1.1) is a nontrivial critical point of J . We follow the argument similar to that of [16], where the problem (1.8) was investigated. Lemma 3.1 (Mountain–Pass lemma, [4]) Let E be a Banach space and ∈ C 1 (E). Assume that (i) (0) = 0. (ii) There exist λ, R > 0 such that (u) ≥ λ for all u ∈ E with u E = R.

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(iii) There exists v0 ∈ E such that lim supt→∞ (tv0 ) < 0. Take t0 > 0 such that t0 v0 E > R and (t0 v0 ) < 0. Set  := {γ ∈ C([0, 1], E)|γ (0) = 0 and γ (1) = t0 v0 }, c := inf sup (γ (t)). γ ∈ t∈[0,1]

Then there exists a Palais–Smale sequence at level c for , that is, there exists a sequence {u k } ⊂ E such that lim (u k ) = c,

k→∞

lim (u k ) = 0 strongly in E −1 .

k→∞

Lemma 3.2 Suppose that (H) holds. Set c∗ := min



 p−N N N −t 1 p−N p−t η1 p S H (μ, 0) p , η2p−t S Q (μ, t) p−t . N p(N − t)

Then for some c ∈ (0, c∗ ), there exists a Palais–Smale sequence at level c for J , that is there exists a sequence {(u k , vk )} ⊂ D × D such that lim J (u k , vk ) = c,

k→∞

lim J (u k , vk ) = 0 strongly in (D × D)−1 .

k→∞

Proof We divide the argument into several steps.

 

Claim 1 The functional J verifies the hypotheses of Lemma 3.1 at any (u, v) ∈ D×D with (u + , v+ ) = (0, 0). In fact, J ∈ C 1 (D × D, R), J (0, 0) = 0. From (1.6) it follows that η1 η2 1 ∗ ∗

(u, v) p −

(u, v) p −

(u, v) p (t) p∗ p∗ (t) p p ∗ S H (μ, 0) p p ∗(t)S Q (μ, t) p

p∗ − p p ∗ (t)− p

(u, v) p , = C1 − C2 (u, v) − C3 (u, v)

J (u, v) ≥

where Ci , i = 1, 2, 3, are positive constants. Then there exist λ, R > 0, such that J (u, v) ≥ λ for all (u, v) ∈ D × D with (u, v) = R. Furthermore, for any (u, v) ∈ D × D with (u + , v+ ) = (0, 0), we have lim J (tu, tv) = −∞,

t→+∞

which implies that there exists t(u,v) > 0 such that (t(u,v) u, t(u,v) v) > R and J (tu, tv) < 0 for all t > t(u,v) . Define (u,v) := {γ ∈ C([0, 1], D × D)|γ (0) = (0, 0) and γ (1) = (t(u,v) u, t(u,v) v)}, c(u,v) :=

123

inf

sup J (γ (t)).

γ ∈(u,v) t∈[0,1]

On the Quasilinear Elliptic Systems Involving Critical

Then the hypotheses of Lemma 3.1 are satisfied and there exists a sequence {(u k , vk )} ⊂ D × D such that lim J (u k , vk ) = c(u,v) ,

k→∞

lim J (u k , vk ) = 0 strongly in (D × D)−1 .

k→∞

In particular, we have that c(u,v) ≥ λ > 0, ∀ (u, v) ∈ D × D \ {(0, 0)}. Claim 2 There exists (u, v) ∈ D × D \ {(0, 0)} such that u, v ≥ 0 and c(u,v) <

N 1 p−N η p S H (μ, 0) p . N 1

 ε (x), In fact, since μ ∈ [0, μ), ¯ by Theorem 1.1 we can choose (u, v) = α1 Vμ,0  ε (x) , the extremals of S (μ, 0). Then β1 Vμ,0 H c(u,v) ≤ sup J (tu, tv) ≤ sup K (t) t≥0 t≥0  p∗ /( p∗ − p)  1

(u, v) p =    ∗ N η1 N H (u, v) p/ p R N 1 p−N = η1 p S H (μ, 0) p , N where K (t) :=

∗ tp tp

(u, v) p − η1 ∗ H (u, v). p p RN

Let t1 , t2 > 0 be the points where supt≥0 J (tu, tv) and supt≥0 K (t) are attained, respectively. Suppose that J (t1 u, t1 v) = K (t2 ). Then p ∗ (t) t1 Q(u, v) K (t1 ) − η2 ∗ = K (t2 ), p (t) R N |x|t

which implies that K (t2 ) < K (t1 ), a contradiction with the definition of t2 . Consequently, c(u,v) ≤ sup J (tu, tv) < sup K (t) = t≥0

t≥0

N 1 p−N η1 p S H (μ, 0) p . N

Claim 3 There exists (u, v) ∈ D × D \ {(0, 0)} such that u, v ≥ 0 and 0 < c(u,v) < c∗ .

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  ε (x), β V ε (x) > 0, the In fact, by Theorem 1.1 we can choose (u, v) = α2 Vμ,t 2 μ,t extremals of S Q (μ, t). Then arguing as above we can obtain that c(u,v) ≤ sup J (tu, tv) t≥0

< sup

t p

(u, v) p − η2

∗ t p (t) Q(u, v)  p ∗ (t) R N |x|t

p p−N N −t p−t η2p−t S Q (μ, t) p−t , = p(N − t) t≥0

which together with claim 2 implies that claim 3 holds. From Lemma 3.1 and claims 1–3 it follows the conclusions of Lemma 3.2 for a suitable (u, v) ∈ D × D. Lemma 3.3 Let {(u k , vk )} ⊂ D × D be a Palais–Smale sequence at the level c < c∗ as in Lemma 3.2. If u k  0 and vk  0 weakly in D as k → ∞, then there exists ε0 > 0 such that for all δ > 0, either lim

k→∞ Bδ (0)

  H (u k )+ , (vk )+ = 0 or

lim

k→∞ Bδ (0)

  H (u k )+ , (vk )+ ≥ ε0 .  

Proof The argument needs several steps. Claim 4 For all  ⊂⊂ R N \ {0}, up to a subsequence, we have |u k | p |vk | p Q(|u k |, |vk |) = lim = lim = 0, lim p p k→∞  |x| k→∞  |x| k→∞  |x|t lim |∇u k | p = lim |∇vk | p = lim H (|u k |, |vk |) = 0.

k→∞ 

k→∞ 

k→∞ 

(3.1) (3.2)

In fact, since  ⊂⊂ R N \ {0}, the embedding D → L q () is compact for any 1 ≤ q < p ∗ , |x|−1 is bounded on  and p ∗ (t) < p ∗ . Then (3.1) follows from (H ) and we only need to verify (3.2). Arguing as in Proposition 2 of [18], take ϕ ∈ C0∞ (R N \ {0}) such that 0 ≤ ϕ ≤ 1 and ϕ| ≡ 1. Note that the weak convergence of {u k } and {vk } in D implies the boundedness. Then

p−1

RN





|∇u k | p−1 |∇(ϕ p )||u k | ≤ ∇u k p

p−1

RN RN

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u k L p (supp|∇ϕ|) = o(1),

|∇vk | p−1 |∇(ϕ p )||vk | ≤ ∇vk p vk L p (supp|∇ϕ|) = o(1),



 p p |ϕ∇u k | + |ϕ∇vk | = |∇(ϕu k )| p + |∇(ϕvk )| p + o(1). RN

On the Quasilinear Elliptic Systems Involving Critical

Furthermore, p p o(1) = J (u k , vk ), (ϕ u k , ϕ vk )  = |ϕ∇u k | p + |ϕ∇vk | p − η1 ϕ p H ((u k )+ , (vk )+ ) N N R R  +O (|∇u k | p−1 |∇(ϕ p )||u k | + |∇vk | p−1 |∇(ϕ p )||vk |) + o(1) RN  p p = |ϕ∇u k | + |ϕ∇vk | − η1 ϕ p (H ((u k )+ , (vk )+ ) + o(1) N R N

R  = |∇(ϕu k )| p + |∇(ϕvk )| p − η1 ϕ p (H ((u k )+ , (vk )+ ) + o(1) N N R R ≥ ϕu k p + ϕvk p − η1 ϕ p H ((u k )+ , (vk )+ ) + o(1),

RN

which implies that

ϕu k p + ϕvk p ≤ η1

ϕ p H ((u k )+ , (vk )+ ) + o(1) ( p∗ − p)/ p∗  p/ p∗ ≤ η1 H ((u k )+ , (vk )+ ) H (|ϕu k |, |ϕvk |) + o(1) N N

R ( p∗ − p)/ p∗ R ≤ η1 H ((u k )+ , (vk )+ ) S H (μ, 0)−1 (ϕu k , ϕvk ) p + o(1), N

R

RN

and therefore, 

1 − η1

RN

H ((u k )+ , (vk )+ )

( p∗ − p)/ p∗

S H (μ, 0)

−1



(ϕu k , ϕvk ) p ≤ o(1). (3.3)

On the other hand, J (u k , vk ) −

1 J (u k , vk ), (u k , vk ) = c + o(1) (u k , vk ) = c + o(1), p

which implies that 1  − ∗ H ((u k )+ , (vk )+ ) c + o(1) = η1 p p RN

1 1  Q((u k )+ , (vk )+ ) − ∗ . + η2 p p (t) R N |x|t

1

Consequently, η1

RN

H ((u k )+ , (vk )+ ) ≤ cN + o(1),

(3.4)

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D. Kang, Y. Kang

which together with (3.3) implies that

 N−p 1 − η1 N (cN ) p/N S H (μ, 0)−1 (ϕu k , ϕvk ) p ≤ o(1). Since c < c∗ , we have that lim (ϕu k , ϕvk ) p = 0,

k→∞

and therefore, lim

k→∞ R N

H (|ϕu k |, |ϕvk |) = 0.

Then the definition of ϕ implies that (3.2) holds and claim 4 is proved. Claim 5 For all δ > 0, define the quantities: Q((u k )+ , (vk )+ ) τ = lim sup H ((u k )+ , (vk )+ ), ω = lim sup , |x|t Bδ (0) Bδ (0) k→∞ k→∞

|u k | p + |vk | p  |∇u k | p + |∇vk | p − μ . γ = lim sup |x| p Bδ (0) k→∞ Then S H (μ, 0) τ

p p∗

p

≤ γ , S Q (μ, t) ω p∗ (t) ≤ γ .

(3.5)

Furthermore, γ ≤ η1 τ + η2 ω.

(3.6)

In fact, according to claim 4, τ, ω, and γ are well defined and independent of δ. Take ϕ ∈ C0∞ (R N ) such that 0 ≤ ϕ ≤ 1 and ϕ| Bδ (0) ≡ 1. Then we have S H (μ, 0)

RN

H ((φu k )+ , (φvk )+ )



p p∗

≤ (ϕu k , ϕvk ) p .

As k → ∞, claim 4 implies that





p p∗

S H (μ, 0) H ((u k )+ , (vk )+ )

Bδ (0) |u k | p + |vk | p  |∇u k | p + |∇vk | p − μ + o(1). ≤ |x| p Bδ (0) Consequently, S H (μ, 0) τ

123

p p∗

≤ γ.

On the Quasilinear Elliptic Systems Involving Critical

The second inequality in (3.5) can be verified similarly. Since ϕu k , ϕvk ∈ D and limk→∞ J (u k , vk ), (ϕu k , ϕvk ) = 0, by claim 4 and the definitions of τ, ω, and γ , we deduce that γ ≤ η1 τ + η2 ω. Claim 5 is verified. From (3.6) it follows that p p∗

S H (μ, 0) τ

≤ γ ≤ η1 τ + η2 ω,

which implies that τ

p p∗

 S H (μ, 0) − η1 τ

p∗ − p p∗

 ≤ η2 ω.

(3.7)

From (3.4) it follows that p−N p

N

η1 τ ≤ cN < c∗ N < η1

p−N p

S H (μ, 0) p = η1

p∗

S H (μ, 0) p∗ − p .

(3.8)

By (3.7) and (3.8), there exists a constant C1 = C1 (μ, c, η1 , η2 ) > 0 such that τ

p p∗

≤ C1 ω.

(3.9)

Similarly, there exists a positive constant C2 = C2 (μ, c, t, η1 , η2 ) such that p

ω p∗ (t) ≤ C2 τ.

(3.10)

Then it follows from (3.9) and (3.10) that there exists a positive constant ε0 = ε0 (N , p, μ, c, t) such that either τ = ω = 0 or min{τ, ω} ≥ ε0 . The proof of Lemma 3.3 is complete.

4 Existence of Positive Solutions Lemma 4.1 Let {(u k , vk )} be the sequence defined as in Lemma 3.3. Then  := lim sup k→∞

RN

H ((u k )+ , (vk )+ ) > 0.

(4.1)

Proof Arguing by contradiction, we assume that lim

k→∞ R N

H ((u k )+ , (vk )+ ) = 0.

(4.2)

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D. Kang, Y. Kang

Since limk→∞ J (u k , vk ), (u k , vk ) = 0, by (4.1) we have

(u k , vk ) p = η2

RN

Q((u k )+ , (vk )+ ) + o(1), k → ∞. |x|t

Then



Q((u k )+ , (vk )+ )  p∗ (t) t |x| RN Q((u k )+ , (vk )+ ) p ≤ (u k , vk ) = η2 + o(1), |x|t RN p

Q((u k )+ , (vk )+ )  p∗ (t) N |x|t R p∗ (t)− p 

Q((u k )+ , (vk )+ )  p∗ (t) ≤ o(1). × S Q (μ, t) − η2 |x|t RN

S Q (μ, t)

p

(4.3)

From (3.4) and (4.2) it follows that η2

c∗ p(N − t) Q((u k )+ , (vk )+ ) cp(N − t) + o(1) < + o(1), = t |x| p−t p−t RN

which together with (4.3) implies that lim

k→∞

Q((u k )+ , (vk )+ ) = 0, |x|t RN

a contradiction with (3.4) and the fact that c ∈ (0, c∗ ).

 

Lemma 4.2 Let {(u k , vk )} be defined as in Lemma 3.3. Then there exists ε1 ∈ (0, ε0 /2], with ε0 given in Lemma 3.3, such that for all ε ∈ (0, ε1 ), there exists a posi  (N − p)/ p  (N − p)/ p tive sequence {rk } ⊂ R such that {(u˜ k , v˜k )} := rk u k (rk x), rk vk (rk x) ⊂ D × D, is again a Palais–Smale sequence of the type given in Lemma 3.3 and satisfies B1 (0)

H ((u˜ k )+ , (v˜k )+ ) = ε, ∀ k ∈ N.

(4.4)

Proof Let ε0 ,  be defined as in Lemma 3.3 and (4.1), respectively. Set ε1 := min{ε0 /2, } and fix ε ∈ (0, ε1 ). Up to a subsequence (still denoted by {(u k , vk )}), for any k ∈ N, there exists rk > 0 such that Brk (0)

H ((u k )+ , (vk )+ ) = ε, ∀ k ∈ N.

Then the scaling invariance implies that {(u˜ k , v˜k )} satisfies (4.4) and is also a Palais– Smale sequence of the type given in Lemma 3.3.  

123

On the Quasilinear Elliptic Systems Involving Critical

Proof of Theorem 1.2 Since {(u˜ k , v˜k )} satisfies (4.4) and is also a Palais–Smale sequence, we have that C(1 + (u˜ k , v˜k ) )

1 ≥ J (u˜ k , v˜k ) − ∗ J (u˜ k , v˜k ), (u˜ k , v˜k ) p (t)     1 1 1 1 = − H ((u˜ k )+ , (v˜k )+ ) − ∗

(u˜ k , v˜k ) p + η1 p (t)  p∗ p ( t) RN p 1 1 − ∗

(u˜ k , v˜k ) p , ≥ p p (t)

which implies that {(u˜ k , v˜k )} is bounded in D × D. Up to a subsequence, there exists u, ˜ v˜ ∈ D such that u˜ k  u˜ weakly,

v˜k  v˜ weakly, k → ∞.

If u˜ ≡ v˜ ≡ 0, from Lemma 3.3 it follows that either lim H ((u˜ k )+ , (v˜k )+ ) = 0 or lim k→∞ B1 (0)

k→∞ B1 (0)

H ((u˜ k )+ , (v˜k )+ ) ≥ ε0 ,

˜ v) ˜ ≡ (0, 0). Arguing as in [12] which contradicts (4.4) as 0 < ε < ε0 /2. Then (u, (see also [13,31,33]), we deduce that (u, ˜ v) ˜ is a solution of the following problem: ⎧ u p−1 η1 ⎪ ⎪ = ∗ Hu (u + , v+ ) + ⎪ − p u − μ ⎪ ⎨ |x| p p v p−1 η1 ⎪ = ∗ Hv (u + , v+ ) + − p v − μ ⎪ ⎪ |x| p p ⎪ ⎩ (u, v) ∈ D × D.

η2 Q u (u + , v+ ) , p ∗ (t) |x|t η2 Q v (u + , v+ ) , p ∗ (t) |x|t

(4.5)

Set w− = max{−w, 0} for all w ∈ D \ {0}. Multiplying the first equation in (4.5) by u˜ − and the second by v˜− , and integrating, we have that u˜ − = v˜− = 0, which ˜ v) ˜ is a nonnegative nontrivial solution implies that u˜ − = v˜− = 0, and therefore, (u, of (4.5). If u˜ ≡ 0, by (H) and (4.5) we get v˜ ≡ 0. Similarly, v˜ ≡ 0 also implies u˜ ≡ 0. Then u˜ ≡ 0 and v˜ ≡ 0. From the maximum principle it follows that u, ˜ v˜ > 0 in R N and (u, ˜ v) ˜ is a solution of the problem (1.1). The proof of Theorem 1.2 is complete.   Acknowledgments This work is supported by the Fundamental Research Funds for the Central Universities, South-Central University for Nationalities (CZW15053).

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