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Solutions of a quasilinear elliptic problem involving a critical Sobolev exponent and multiple Hardy-type terms
J. Math. Anal. Appl. 333 (2007) 889–903 www.elsevier.com/locate/jmaa
Solutions of a quasilinear elliptic problem involving a critical Sobolev exponent and multiple Hardy-type terms ✩ Yuping Cao, Dongsheng Kang ∗ Department of Mathematics, South-Central University for Nationalities, Wuhan 430074, PR China Received 25 January 2006 Available online 10 January 2007 Submitted by J. Lavery
Abstract In the present paper, a quasilinear elliptic problem with a critical Sobolev exponent and multiple Hardytype terms is considered. By means of a variational method, the existence of positive solutions of the problem is obtained. © 2006 Elsevier Inc. All rights reserved. Keywords: Solution; Quasilinear problem; Critical exponent; Variational method
1. Introduction and the main results In this paper, we will study the following quasilinear elliptic problem: ⎧ k ⎪ |u|p−2 ∗ ⎨ −p u − μi u = Q(x)|u|p −2 u + λ|u|q−2 u, x ∈ Ω, |x − ai |p i=1 ⎪ ⎩ u = 0, x ∈ ∂Ω,
(1.1)
where Ω is a bounded domain in RN (N 3) with smooth boundary ∂Ω, −p u = −div (|∇u|p−2 ∇u) is the p-Laplacian of u, λ 0, 1 < p < N , p q < p ∗ , ai ∈ Ω and ✩
This work is supported partly by NSF of China and NSF of South-Central University for Nationalities.
* Corresponding author.
E-mail address: [email protected] (D. Kang). 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.12.005
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0 < μi < μ¯ with i = 1, 2, . . . , k and k 2. μ¯ = (N − p)p /p p is the best Hardy constant, p ∗ = Np/(N − p) is the critical Sobolev exponent and Q(x) is a positive bounded function on Ω. Problem (1.1) is related to the following Hardy inequality [3,11,12,14]: |u|p 1 (1.2) dx |∇u|p dx, ∀a ∈ RN , u ∈ C0∞ RN . p |x − a| μ¯ RN
RN
By taking a = ai (i = 1, 2, . . . , k) in (1.2) we obtain |u|p 1 dx |∇u|p dx, ∀u ∈ C0∞ RN , i = 1, 2, . . . , k. p |x − ai | μ¯ RN
(1.3)
RN
These inequalities will be employed in the following argument. 1,p In this paper, W0 (Ω) denotes the space obtained as the completion of C0∞ (Ω) with respect to the norm
1 p p |∇u| dx . uW 1,p (Ω) = u = 0
Ω
is said to be a solution of problem (1.1) if u satisfies
k |u|p−2 p−2 p ∗ −2 q−2 ∇u∇v − μi uv − |u| uv − λ|u| uv dx = 0 |∇u| |x − ai |p
The function Ω
1,p u ∈ W0 (Ω)
i=1
1,p
for all v ∈ W0 (Ω). By the standard elliptic regularity argument we deduce that the solution u ∈ C 1 (Ω \ {a1 , a2 , . . . , ak }). It is well known that the nontrivial solution of problem (1.1) is equivalent to the corresponding nonzero critical point of the energy functional
k |u|p λ 1 1 p p∗ μi |u| dx − |u|q dx, J (u) = |∇u| − dx − p |x − ai |p p∗ q Ω
i=1
Ω
Ω
1,p defined on the space W0 (Ω). ∗ D 1,p (RN ) = {u ∈ Lp (RN ); |∇u|
which is well ∈ Lp (RN )} is also used. By Hardy inequality The space and Sobolev inequality, the following best Sobolev constant is well defined for every μi ∈ [0, μ), ¯ i = 1, 2, . . . , k: |u|p p RN (|∇u| − μi |x−ai |p ) dx inf . Sμ i = p ∗ u∈D 1,p (RN )\{0} ( RN |u|p dx) p∗ If p q p ∗ , we can also define the following best constant: |u|p p Ω (|∇u| − μi |x−ai |p ) dx Sμi ,q (Ω) = inf . p 1,p u∈W0 (Ω)\{0} ( Ω |u|q dx) q In the important case when q = p ∗ , we denote Sμi ,p∗ (Ω) as Sμi (Ω). Sμi (Ω) is independent of Ω in the sense that Sμi (Ω) = Sμi (RN ) = Sμi .
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By the Hardy inequality (1.3), if μi 0 (i = 1, 2, . . . , k) and defined as L(u) = −p u −
k i=1
μi
|u|p−2 u , |x − ai |p
k
i=1 μi
891
< μ, ¯ the operator L
1,p
u ∈ W0 (Ω)
is positive and the first eigenvalue λ1 of L is well defined, namely, k |u|p p i=1 μi |x−ai |p ) dx Ω (|∇u| − λ1 = λ1 (Ω) = . inf p 1,p u∈W (Ω)\{0} Ω |u| dx 0
Another challenging problem related to (1.1) is to investigate the extremal functions by which the best constant Sμi is achieved. Here we recall the recent work of Abdellaoui, Felli and Peral [1], where the authors studied the following limiting problem: ⎧ p−1 ⎪ ∗ ⎨ − u − μ u = up −1 in RN \ {ai }, p i p |x − ai | ⎪ ⎩ u ∈ D 1,p RN , u > 0 in RN \ {ai }. They proved that for 0 μi < μ¯ and 1 < p < N , the problem has radially symmetric ground states
p−N p−N x − ai |x − ai | ai p p =ε , ∀ε > 0, (1.4) Vp,μi ,ε (x) = ε Up,μi Up,μi ε ε that satisfy
∇V ai RN
ai p p − μi |Vp,μi ,ε (x)| (x) p,μi ,ε |x − ai |p
=
a V i
p ∗ N = (Sμ ) p . i
p,μi ,ε (x)
(1.5)
RN
Moreover, the positive radial functions Up,μi have the following properties: lim r a(μi ) Up,μi (r) = C1 > 0,
(1.6)
lim r b(μi ) Up,μi (r) = C2 > 0, (r) = C1 a(μi ) 0, lim r a(μi )+1 Up,μ i r→0 (r) = C2 b(μi ) > 0, lim r b(μi )+1 Up,μ i
(1.7)
r→0
r→+∞
(1.8) (1.9)
r→+∞
where C1 and C2 are positive constants depending on p and N and a(μi ) and b(μi ) are zeroes of the function f (t) = (p − 1)t p − (N − p)t p−1 + μi ,
t 0,
(1.10)
that satisfy 0 a(μi ) <
N −p < b(μi ), p
i = 1, 2, . . . , k.
The above results are crucial to the study of problem (1.1). For example, in the recent work of Han [13], a particular case of problem (1.1) was studied, the case when k = 1 and q = p. The
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author applied properties (1.4)–(1.10) and obtained the existence of positive solutions by variational arguments. We mention that the singular quasilinear problems related to Hardy inequality were investigated not only in [1] and [13] but also in [11] and [12] and in the references therein. Now we recall another important singular problem, to which much attention has been paid in the past several years, namely, u ∗ −u − μ 2 = |u|2 −2 u + λ|u|q−2 u, x ∈ Ω, |x| (1.11) u = 0, x ∈ ∂Ω, where Ω is a smooth bounded domain in RN (N 3), 0 ∈ Ω, λ > 0, 0 μ < (N − 2)2 /4, 2 q < 2∗ and 2∗ = 2N/(N − 2) is the critical Sobolev exponent. This problem was studied in [4,6–8,10,15–17,20,22] and in the references therein. Many important results on the nontrivial solutions of (1.11) were obtained in these publications and these results give us very good insight into the problem. Most recently, the authors of [5] and [9] studied the following semilinear problem with multiple Hardy-type terms: ⎧ k ⎪ μi ∗ ⎨ u = |u|2 −2 u, x ∈ Ω, −u − 2 (1.12) |x − ai | i=1 ⎪ ⎩ u = 0, x ∈ ∂Ω, where Ω is also a bounded domain in RN with smooth boundary, N 3, ai ∈ Ω and 0 μi < (N − 2)2 /4 for i = 1, 2, . . . , k and k 2. By applying the Hardy inequality and critical point theory, some elegant results about problem (1.12) were obtained, such as the existence and multiplicity of solutions, the asymptotic behaviors of solutions and so on. These results enable us to understand problem (1.11) better. Therefore, an interesting question arises naturally, that is, whether the results about the semilinear equation (1.12) remain true for (1.1), the quasilinear problem with a critical Sobolev exponent and multiple Hardy-type terms? Stimulated by [1,5,9,13], in this paper we shall study the existence of solutions of problem (1.1) and give some positive answers to the above question. To our knowledge, there are no results on the existence of nontrivial solutions for (1.1) for k 2. It is therefore meaningful for us to investigate problem (1.1) deeply. However, due to the singularities caused by the terms |x − ai |−p (i = 1, 2, . . . , k), the quasilinear problem (1.1) becomes more complicated to deal with than (1.12) and therefore we have to face more difficulties. Now we recall the following standard definition. Assume that X is a Banach space, X −1 is the dual space of X. The functional I ∈ C 1 (X, R) is said to satisfy the Palais–Smale condition at level c ((PS)c in short), if every sequence {un } ⊂ X satisfying I (un ) → c and I (un ) → 0 in X −1 has a convergent subsequence. 1,p In this paper, we will take I = J and X = W0 (Ω). To proceed, we need the following assumptions: (H1 ) There exists an l ∈ {1, 2, . . . , k} such that (Sμl )
N p
N p−N p−N p Q(al ) p = min (Sμi ) Q(ai ) p , i = 1, 2, . . . , k .
(H2 ) Q(x) is a positive bounded function on Ω and there exists an x0 ∈ Ω such that Q(x0 ) is a strict local maximum. Furthermore,
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Q(x0 ) = QM = max Q(x), Ω Q(x) − Q(x0 ) = o |x − x0 |p as x → x0 and
Q(x) − Q(al ) = o |x − al |p as x → al . ¯ (H3 ) 0 < μi < μ¯ for every i = 1, 2, . . . , k and ki=1 μi < μ. Now we are ready to state the main results of this paper, which are concluded as the following theorem. By the fact that b(μl ) > (N − p)p −1 we can verify that the intervals for q and μl in the following theorem are allowable. To the best of our knowledge, the results are new in the case when k 2 and p = 2. Theorem 1.1. Suppose (H1 ), (H2 ) and (H3 ). Assume that one of the following conditions holds: (i) λ > 0 and q¯ < q < p ∗ , where p(2N − b(μl )p − p) N , . q¯ = max p, b(μl ) N −p (ii) N > p 2 , q = p, 0 < λ < λ1 and 0 < μl (N − p 2 )N p−1 p −p . (iii) N > p 2 , p < q < p ∗ , λ 0 and 0 < μl (N − p 2 )N p−1 p −p . 1,p
Then problem (1.1) has at least one positive solution u ∈ W0 (Ω). The proof of Theorem 1.1 will be given in Section 2. At the end of this section, we explain 1,p 1,p some notations employed in this paper: (W0 (Ω))−1 is the dual space of W0 (Ω), O(ε t ) denotes the quantity satisfying |O(ε t )|/ε t C and o(ε t ) means |o(ε t )|/ε t → 0 as ε → 0. In the following argument, we employ C to denote the positive constant and omit dx in integral for convenience. 2. Proof of the main results In this section, we will prove Theorem 1.1 by employing the mountain pass theorem due to Ambrosetti and Rabinowitz [2]. For this purpose, we need to establish several preliminary lemmas. Lemma 2.1. The functional J satisfies (PS)c condition with N N p−N p−N 1 p p c < c∗ = min (Sμl ) Q(al ) p , (S0 ) (QM ) p . N Proof. The argument is similar to that of Lemma 2.1 in [13] and a suitable version of the concentration compactness principle [18,19] will be employed, here we omit the details. 2 In the following, we investigate the extremal function defined in (1.4). To this end, we define the function
p−N x − ai ai ai p , uμi ,ε (x) = ϕai (x)Vp,μi ,ε (x) = ε ϕai (x)Up,μi ε
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where the cut-off function ϕai ∈ C0∞ (Bρ (ai )), 0 ϕai 1 and ϕai ≡ 1 for all x ∈ Bρ/2 (ai ). The radius ρ is chosen small enough such that 0 < ρ < min |ai − aj |, i, j = 1, 2, . . . , k, i = j and Bρ (ai ) ⊂ Ω. Then there exist positive constants C1 and C2 such that C1 |x − aj | C2 ,
∀x ∈ Bρ (ai ), 1 j k, j = i.
ai If ε → 0, the behavior of uaμii ,ε (x) has to be the same as that of the extremal function Vp,μ i ,ε (x), but we need the precise estimates of the error terms. To this end, setting
βi = b(μi ) −
N −p , p
i = 1, 2, . . . , k,
then we have the following estimates (2.1)–(2.4). The proof of Lemma 2.2 follows the same line as that of [13], here we omit it. Lemma 2.2. For ε > 0 small and any i, j = 1, 2, . . . , k with i = j, we have the following estimates:
ai p a p N ∇u i − μi |uμi ,ε | = (Sμi ) p + O ε pβi , (2.1) μi ,ε |x − ai |p Ω a p ∗ N u i = (Sμ ) p + O ε p∗ βi , (2.2) μi ,ε i Ω
⎧ N +(1− N N ∗ ⎪ p )q , Cε ⎪ ⎪ b(μi ) < q < p , ⎨ a q u i N +(1− N N p )q |ln ε|, Cε q = b(μ , μi ,ε ⎪ i) ⎪ ⎪ Ω ⎩ Cε qβi , N p q < b(μ , i) ⎧ Cε p , b(μi ) > Np , ⎪ ⎪ ⎨ |uaμii ,ε |p Cε p |ln ε|, b(μi ) = Np , |x − aj |p ⎪ ⎪ ⎩ Cε pβi , Ω b(μ ) < N . i
(2.3)
(2.4)
p
Furthermore, as ε → 0 we have a q u i → 0, p q < p ∗ μi ,ε Ω
and
Ω
|uaμii ,ε |p → 0, |x − aj |p
i, j = 1, 2, . . . , k with i = j.
For further discussion, we recall the following result of Talenti [21]: For all ε > 0, a ∈ RN and N−p CN,p = N(N − p)(p − 1)−1 p2 ,
Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
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the best Sobolev constant S0 is achieved by the extremal functions p−N p p−N a = CN,p ε p 1 + |x − a| p−1 p , Vp,0,ε that satisfy ∗ N ∇V a p = V a p = (S0 ) p . p,0,ε p,0,ε RN
RN
a We deal with the function Vp,0,ε by the method similar to that of Lemma 2.2. To this end, for all a ∈ Ω \ {a1 , a2 , . . . , ak } we define the function a ua0,ε (x) = ϕ(x)Vp,0,ε (x),
where the cut-off function ϕ(x) ∈ C0∞ (Bρ (a)), 0 ϕ(x) 1 and ϕ(x) ≡ 1 for all x ∈ Bρ/2 (a). The radius ρ is chosen small enough such that 0 < ρ < min |a − ai |, i = 1, 2, . . . , k and Bρ (a) ⊂ Ω. Then there exist positive constants C1 and C2 such that C1 |x − ai | C2 ,
∀x ∈ Bρ (a), i = 1, 2, . . . , k.
Furthermore, by taking μi = 0 in (2.1)–(2.4) and from the fact that b(0) = (N − p)/(p − 1) we have the following lemma. Lemma 2.3. For ε > 0 small and every i ∈ {1, 2, . . . , k} we have the following estimates: a p N N−p ∇u = (S0 ) p + O ε p−1 , (2.5) 0,ε Ω
a p ∗ N N u = (S0 ) p + O ε p−1 , 0,ε
(2.6)
Ω
⎧ N +(1− N )q N (p−1) ∗ p ⎪ Cε , ⎪ N −p < q < p , ⎪ ⎨ a q N u Cε N +(1− p )q |ln ε|, q = N (p−1) , 0,ε N −p ⎪ ⎪ ⎪ Ω ⎩ q(N−p) Cε p(p−1) , p q < NN(p−1) −p , ⎧ p 2 N >p , ⎪ ⎨ Cε , |ua0,ε |p p Cε |ln ε|, N = p 2 , |x − ai |p ⎪ ⎩ N−p Ω Cε p−1 , N < p2 . Furthermore, as ε → 0 we have a q u → 0, p q < p ∗ 0,ε Ω
and
Ω
|ua0,ε |p |x − ai |p
→ 0.
(2.7)
(2.8)
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Now we are in the position to verify the properties of the functional J . The argument is based on the analytical techniques and the results of Lemmas 2.2 and 2.3. Lemma 2.4. Suppose (H1 ), (H2 ) and (H3 ). Assume that one of the following conditions holds: (i) λ > 0 and q¯ < q < p ∗ , where p(2N − b(μl )p − p) N , q¯ = max p, . b(μl ) N −p (ii) N > p 2 , q = p, 0 < λ < λ1 and 0 < μl (N − p 2 )N p−1 p −p . (iii) N > p 2 , p < q < p ∗ , λ 0 and 0 < μl (N − p 2 )N p−1 p −p . 1,p
Then there exists a nonnegative function v ∈ W0 (Ω) \ {0} such that sup J (tv) < c∗ , t0
where c∗ is defined as in Lemma 2.1. Proof. To prove the lemma, we need to distinguish the following cases: N p
Case I. (Sμl ) (Q(al )) and N p
p−N p
N p
< (S0 ) (QM ) N p
p−N
p−N p
p−N
Case II. (Sμl ) (Q(al )) p (S0 ) (QM ) p . We first study Case I. The definition of c∗ implies that c∗ =
N p−N 1 p (Sμl ) Q(al ) p . N
Then the function [0, +∞): tp g(t) ¯ = p Ω
a
uμll ,ε in Lemma 2.2 is used to define the following functions on the interval
∗ al p p ∗ a p tp ∇u l − μl |uμl ,ε | − Q(x)uaμll ,ε μl ,ε p ∗ |x − al | p Ω
and g(t) = J tuaμll ,ε = g(t) ¯ −
k |uaμll ,ε |p tp t q al q uμl ,ε . μi − λ p |x − ai |p q i =l,i=1
Ω
Ω
For ρ > 0 small and η > 0, it follows from (H2 ) that ∗ ∗ ∗ Q(x)ual p − Q(al )ual p Q(x) − Q(al )ual p μl ,ε μl ,ε μl ,ε Ω
Ω
Ω
Cη
p ∗ |x − al |p uaμll ,ε
Ω
Cηε p .
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By (2.2) we have p∗ N ∗ Q(x)uaμll ,ε = Q(al )(Sμl ) p + O ε p βl + o ε p . Ω
Then from the fact that
p ∗ t 1 N p−N tp sup B1 − ∗ B2 = B1p B2 p , p N t0 p
B1 > 0, B2 > 0
we obtain 1 sup g(t) ¯ = N t0
N p−N
al p p a p ∗ p a p l ∇u l − μl |uμl ,ε | Q(x) uμl ,ε μl ,ε |x − al |p
Ω
Ω
N N N p−N ∗ 1 (Sμl ) p + O ε pβl p Q(al )(Sμl ) p + O ε p βl + o ε p p = N N p−N 1 (Sμl ) p Q(al ) p + O ε pβl + o ε p . N
Furthermore, the fact that limt→+∞ g(t) = −∞ and g(t) > 0 as t → 0+ implies supt0 g(t) must be attained at some finite tε > 0. Then from g (tε ) = 0 we deduce that
al p a p ∇u l − μl |uμl ,ε | μl ,ε |x − al |p Ω
= tεp
∗ −p
p∗ Q(x)uaμll ,ε + λtεq−p
Ω
k a q u l + μ i μl ,ε i =l,i=1
Ω
Ω
|uaμll ,ε |p . |x − ai |p
Hence, we obtain from Lemma 2.2 that (2.9)
C1 < tε < C2 , where C1 and C2 are positive constants independent of ε. To continue, we consider the following cases. (i) λ > 0 and q¯ < q < p ∗ , where p(2N − b(μl )p − p) N , . q¯ = max p, b(μl ) N −p In this case, from Lemma 2.2 we have a |uμll ,ε |p lim = 0, i = 1, 2, . . . , k, i = l. ε→0 |x − ai |p Ω
Consequently, p
g(tε ) = g(t ¯ ε) −
tε p
k i =l,i=1
μi Ω
q
|uaμll ,ε |p tε −λ p |x − ai | q
a q u l μl ,ε
Ω
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Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903 q
sup g(t) ¯ −λ t0
tε q
a q u l μl ,ε
Ω
N p−N 1 (Sμl ) p Q(al ) p + O ε pβl + o ε p − C N
a q u l . μl ,ε
Ω
From (2.3) and the assumption q¯ < q a q N u l Cε N +(1− p )q μl ,ε
< p∗
we obtain
Ω
and
N q < min{p, pβl }. N + 1− p
Taking ε small enough we have g(tε ) <
N p−N 1 (Sμl ) p Q(al ) p . N
(ii) q = p, 0 < λ < λ1 and b(μ)
(2.10)
N p.
The same argument as that in case (i) shows that λ p al p uμl ,ε ¯ ε ) − tε g(tε ) g(t p Ω N p−N 1 (Sμl ) p Q(al ) p + O ε pβl + o ε p − C N
Ω
If b(μl ) > Np , from (2.3) we obtain a p u l Cε p . μl ,ε Ω
Then (2.10) follows by taking ε small enough. If b(μl ) = Np , from the fact a p u l Cε p |ln ε| μl ,ε Ω
we obtain that (2.10) holds for ε small. (iii) N > p 2 , λ 0 and p < q < p ∗ . In this case, from Lemma 2.2 we obtain p
¯ ε) − g(tε ) g(t
tε p
k i =l,i=1
μi Ω
|uaμll ,ε |p |x − ai |p
a p u l . μl ,ε
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k N p−N |uaμll ,ε |p 1 . (Sμl ) p Q(al ) p + O ε pβl + o ε p − C N |x − ai |p
i =l,i=1 Ω
If b(μl ) >
N p,
k i =l,i=1 Ω
(2.4) implies that
|uaμll ,ε |p Cε p . |x − ai |p
Hence, (2.10) holds naturally for ε small. If b(μl ) = Np , from (2.4) we obtain k i =l,i=1 Ω
|uaμll ,ε |p Cε p |ln ε|. |x − ai |p
Thus (2.10) follows by taking ε small enough. On the other hand, it is easy to verify that the function f (t) = (p − 1)t p − (N − p)t p−1 + μl ,
t ∈ [0, +∞)
has the only minimal point t¯ = (N − p)/p and is increasing on the interval (t¯, +∞). Thus for N > p 2 we have
N N N p−1 (N − p 2 ) b(μl ) ⇐⇒ f f b(μl ) = 0 ⇐⇒ 0 < μl . p p pp Hence Case I is verified. Next, we investigate Case II. In this case we have p−N N N N p−N p−N 1 1 1 c∗ = (S0 ) p (QM ) p = (S0 ) p Q(x0 ) p (Sμl ) p Q(al ) p , N N N where x0 is the maximum point of Q(x) defined as in (H2 ). If x0 = ai for some i ∈ {1, 2, . . . , k}, from the fact that Sμi < S0 we obtain N N N p−N p−N p−N 1 1 1 (S0 ) p Q(ai ) p > (Sμi ) p Q(ai ) p (Sμl ) p Q(al ) p , N N N which is impossible. Hence x0 = ai for any i ∈ {1, 2, . . . , k}. x0 In the following argument, we employ the function u0,ε given in Lemma 2.3. Consider the functions defined on the interval [0, +∞): p p∗ x0 p∗ ∇ux0 p − t ¯h(t) = t Q(x)u0,ε 0,ε p p∗
c∗ =
Ω
Ω
and x0 p k p |u0,ε | x0 t q x0 q ¯ −t u0,ε . h(t) = J tu0,ε = h(t) μi − λ p |x − ai |p q i=1
Then from (2.5)–(2.8) and (H2 ) we have
Ω
Ω
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Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
¯ = 1 sup h(t) N t0
x0 p ∇u
N p
0,ε
Ω
x0 p∗ Q(x)u0,ε
− N−p p
Ω
N N − N−p N−p N N 1 p (S0 ) p + O ε p−1 p Q(x0 )(S0 ) p + O ε p−1 + o ε p = N N p−N N−p 1 (S0 ) p Q(x0 ) p + O ε p−1 + o ε p . N Arguing as in Case I, we have that supt0 h(t) must be attained at some tε and there exist positive constants C1 and C2 such that C1 < tε < C2 . Furthermore,
¯ − h(tε ) sup h(t) t0
x0 p p k q |u0,ε | tε x0 q tε u0,ε μi − λ p |x − ai |p q i=1
Ω
Ω
N p−N N−p 1 (S0 ) p Q(x0 ) p + O ε p−1 + o ε p N x0 p k |u0,ε | x0 q u . μi − C −C 0,ε |x − ai |p
i=1
Ω
(2.11)
Ω
To proceed, we need to investigate the following cases. (i) λ > 0 and q¯0 < q < p ∗ , where N (p − 1) ∗ p ,p − . q¯0 = max p, N −p p−1 From Lemma 2.3 we have x0 p |u0,ε | lim = 0, i = 1, 2, . . . , k. ε→0 |x − ai |p Ω
Then (2.11) implies N p−N N−p 1 h(tε ) (S0 ) p Q(x0 ) p + O ε p−1 + o ε p − C N
x0 q u . 0,ε
Ω
Furthermore, from (2.7) and the assumption q¯0 < q a q N u l Cε N +(1− p )q μl ,ε
< p∗
we deduce that
Ω
and
N N −p N + 1− q < min p, . p p−1
Consequently, N p−N 1 (S0 ) p Q(x0 ) p N for ε small enough.
h(tε ) <
(2.12)
Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
901
On the other hand, from the equality (p − 1)b(μl )p − (N − p)b(μl )p−1 + μl = 0 we obtain
(p − 1)b(μl )
p−1
N −p b(μl ) − p−1
+ μl = 0.
Then the following inequalities N −p N −p < b(μl ) < p p−1 and N p(2N − b(μl )p − p) p N(p − 1) ∗ ,p − max p, , max p, N −p p−1 b(μl ) N −p hold for all μl ∈ (0, μ). ¯ Furthermore, q¯0 q¯
and (q, ¯ p ∗ ) ⊂ (q¯0 , p ∗ ),
where the constant q¯ is defined as in Case I. Hence q ∈ (q¯0 , p ∗ ) under the stronger hypothesis q ∈ (q, ¯ p ∗ ) of Case I. (ii) N > p 2 , q = p and 0 < λ < λ1 . N −p p−1 .
Then (2.7) and (2.11) imply that x0 p N p−N N−p 1 h(tε ) (S0 ) p Q(x0 ) p + O ε p−1 + o ε p − C u0,ε N
In this case we have p <
Ω
N p−N N−p 1 (S0 ) p Q(x0 ) p + O ε p−1 + o ε p − Cε p . N Thus (2.12) holds naturally for ε small.
(iii) N > p 2 , λ 0 and p < q < p ∗ . From (2.8) and (2.11) and by the fact that p < h(tε )
N −p p−1
we obtain
k N p−N N−p 1 (S0 ) p Q(x0 ) p + O ε p−1 + o ε p − C N
i=1 Ω
x
0 p |u0,ε |
|x − ai |p
N p−N N−p 1 (S0 ) p Q(x0 ) p + O ε p−1 + o ε p − Cε p . N Then (2.12) follows by taking ε small enough. Hence, Case II is proved. From Cases I and II we conclude Lemma 2.4. 2
Proof of Theorem 1.1. We verify that the functional J satisfies the mountain pass geometry. To this end, we consider the energy level c = inf max J γ (t) , γ ∈Γ t∈[0,1]
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Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
where 1,p Γ = γ ∈ C [0, 1], W0 (Ω) γ (0) = 0, J γ (1) < 0 . If λ 0 and p < q < p ∗ , Hardy inequality and Sobolev inequality imply that
k μi |u|p λ 1 p p∗ |u|q |∇u| − − ∗ Q(x)|u| − |x − ai |p p q
1 J (u) = p
i=1
Ω
1 1 μi 1− p μ¯ k
i=1
−
λ −q (S0,q ) p q
Ω
Ω
∗ QM −p |∇u| − ∗ (S0 ) p p
Ω
p
p
Ω
q
|∇u|p
p∗ |∇u|
p
p
.
Ω
If 0 < λ < λ1 and q = p, then by the definition of λ1 we have that 1 J (u) = p
k μi |u|p λ 1 p p∗ Q(x)|u| − |u|p |∇u| − − |x − ai |p p∗ p i=1
Ω
Ω
Ω
p∗ k p p∗ λ 1 Q 1 M − 1− μi |∇u|p − ∗ (S0 ) p |∇u|p . 1− p λ1 μ¯ p
i=1
Ω
Ω
Hence, there exists ρ¯ > 0 small enough such that b = inf J (u) > 0 = J (0). u=ρ¯
1,p
Now we employ the function v ∈ W0 (Ω) \ {0} obtained in Lemma 2.4. Then there exists t0 > 0 such that t0 v > ρ¯ and J (t0 v) < 0. By the mountain pass theorem [2], there exists a 1,p sequence {un } ⊂ W0 (Ω) such that J (un ) → 0
J (un ) → c,
1,p −1 in W0 (Ω) .
From Lemma 2.4 we obtain c sup J (tt0 v) sup J (tv) < c∗ . t∈[0,1]
t0
According to Lemma 2.1, there exists a subsequence of {un } (still denoted by {un }), such that 1,p un → u strongly in W0 (Ω). Then u is a critical point of the functional J and thereby a solution of problem (1.1), c is the corresponding critical value of J . In order to find the positive solution of (1.1), we replace the functional J (u) with the following J + (u):
k + p∗ λ + q μi |u|p 1 1 + p u Q(x) u − , J (u) = |∇u| − − p |x − ai |p p∗ q Ω
i=1
Ω
Ω
Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
903
where u+ = max{u, 0} ∈ W0 (Ω). Repeating the above argument, we obtain a critical point u of J + , which satisfies the following equation ⎧ k ⎪ p∗ −1 q−1 |u|p−2 u ⎨ −p u − μi = Q(x) u+ + λ u+ , x ∈ Ω, p |x − ai | ⎪ i=1 ⎩ u = 0, x ∈ ∂Ω. 1,p
By the maximum principle [23], we have that u > 0 in Ω. The proof of Theorem 1.1 is therefore completed. 2 References [1] B. Abdellaoui, V. Felli, I. Peral, Existence and nonexistence for quasilinear equations involving the p-Laplacian, preprint. [2] A. Ambrosetti, H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [3] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequality with weights, Compos. Math. 53 (1984) 259–275. [4] D. Cao, P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations 205 (2004) 521–537. [5] D. Cao, P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differential Equations 224 (2006) 332–372. [6] D. Cao, S. Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms, J. Differential Equations 193 (2003) 424–434. [7] J. Chen, Multiple solutions for a class of nonlinear elliptic equations, J. Math. Anal. Appl. 295 (2004) 341–354. [8] I. Ekeland, N. Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc. 39 (2002) 207–265. [9] V. Felli, S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations 31 (2006) 469–495. [10] A. Ferrero, F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations 177 (2001) 494–522. [11] J.P. Garcia Azorero, I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998) 441–476. [12] N. Ghoussoub, C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000) 5703–5743. [13] P. Han, Quasilinear elliptic problems with critical exponents and Hardy terms, Nonlinear Anal. 61 (2005) 735–758. [14] G. Hardy, J. Littlewood, G. Polya, Inequalities, Cambridge Univ. Press, Cambridge, UK, 1934. [15] E. Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations 156 (1999) 407–426. [16] D. Kang, Y. Deng, Existence of solution for a singular critical elliptic equation, J. Math. Anal. Appl. 284 (2003) 724–732. [17] D. Kang, S. Peng, Existence of solutions for elliptic equations with critical Sobolev–Hardy exponents, Nonlinear Anal. 56 (2004) 1151–1164. [18] P.L. Lions, The concentration compactness principle in the calculus of variations, the limit case (I), Rev. Math. Iberoamericana 1 (1) (1985) 145–201. [19] P.L. Lions, The concentration compactness principle in the calculus of variations, the limit case (II), Rev. Math. Iberoamericana 1 (2) (1985) 45–121. [20] D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc. 357 (2005) 2909–2938. [21] G. Talenti, Best constants in Sobolev inequalities, Ann. Mat. Pura Appl. 110 (1976) 353–372. [22] S. Terracini, On positive solutions to a class equations with a singular coefficient and critical exponent, Adv. Differential Equations 2 (1996) 241–264. [23] J.L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984) 191–202.
Solutions of a quasilinear elliptic problem involving a critical Sobolev exponent and multiple Hardy-type terms ✩ Yuping Cao, Dongsheng Kang ∗ Department of Mathematics, South-Central University for Nationalities, Wuhan 430074, PR China Received 25 January 2006 Available online 10 January 2007 Submitted by J. Lavery
Abstract In the present paper, a quasilinear elliptic problem with a critical Sobolev exponent and multiple Hardytype terms is considered. By means of a variational method, the existence of positive solutions of the problem is obtained. © 2006 Elsevier Inc. All rights reserved. Keywords: Solution; Quasilinear problem; Critical exponent; Variational method
1. Introduction and the main results In this paper, we will study the following quasilinear elliptic problem: ⎧ k ⎪ |u|p−2 ∗ ⎨ −p u − μi u = Q(x)|u|p −2 u + λ|u|q−2 u, x ∈ Ω, |x − ai |p i=1 ⎪ ⎩ u = 0, x ∈ ∂Ω,
(1.1)
where Ω is a bounded domain in RN (N 3) with smooth boundary ∂Ω, −p u = −div (|∇u|p−2 ∇u) is the p-Laplacian of u, λ 0, 1 < p < N , p q < p ∗ , ai ∈ Ω and ✩
This work is supported partly by NSF of China and NSF of South-Central University for Nationalities.
* Corresponding author.
E-mail address: [email protected] (D. Kang). 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.12.005
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Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
0 < μi < μ¯ with i = 1, 2, . . . , k and k 2. μ¯ = (N − p)p /p p is the best Hardy constant, p ∗ = Np/(N − p) is the critical Sobolev exponent and Q(x) is a positive bounded function on Ω. Problem (1.1) is related to the following Hardy inequality [3,11,12,14]: |u|p 1 (1.2) dx |∇u|p dx, ∀a ∈ RN , u ∈ C0∞ RN . p |x − a| μ¯ RN
RN
By taking a = ai (i = 1, 2, . . . , k) in (1.2) we obtain |u|p 1 dx |∇u|p dx, ∀u ∈ C0∞ RN , i = 1, 2, . . . , k. p |x − ai | μ¯ RN
(1.3)
RN
These inequalities will be employed in the following argument. 1,p In this paper, W0 (Ω) denotes the space obtained as the completion of C0∞ (Ω) with respect to the norm
1 p p |∇u| dx . uW 1,p (Ω) = u = 0
Ω
is said to be a solution of problem (1.1) if u satisfies
k |u|p−2 p−2 p ∗ −2 q−2 ∇u∇v − μi uv − |u| uv − λ|u| uv dx = 0 |∇u| |x − ai |p
The function Ω
1,p u ∈ W0 (Ω)
i=1
1,p
for all v ∈ W0 (Ω). By the standard elliptic regularity argument we deduce that the solution u ∈ C 1 (Ω \ {a1 , a2 , . . . , ak }). It is well known that the nontrivial solution of problem (1.1) is equivalent to the corresponding nonzero critical point of the energy functional
k |u|p λ 1 1 p p∗ μi |u| dx − |u|q dx, J (u) = |∇u| − dx − p |x − ai |p p∗ q Ω
i=1
Ω
Ω
1,p defined on the space W0 (Ω). ∗ D 1,p (RN ) = {u ∈ Lp (RN ); |∇u|
which is well ∈ Lp (RN )} is also used. By Hardy inequality The space and Sobolev inequality, the following best Sobolev constant is well defined for every μi ∈ [0, μ), ¯ i = 1, 2, . . . , k: |u|p p RN (|∇u| − μi |x−ai |p ) dx inf . Sμ i = p ∗ u∈D 1,p (RN )\{0} ( RN |u|p dx) p∗ If p q p ∗ , we can also define the following best constant: |u|p p Ω (|∇u| − μi |x−ai |p ) dx Sμi ,q (Ω) = inf . p 1,p u∈W0 (Ω)\{0} ( Ω |u|q dx) q In the important case when q = p ∗ , we denote Sμi ,p∗ (Ω) as Sμi (Ω). Sμi (Ω) is independent of Ω in the sense that Sμi (Ω) = Sμi (RN ) = Sμi .
Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
By the Hardy inequality (1.3), if μi 0 (i = 1, 2, . . . , k) and defined as L(u) = −p u −
k i=1
μi
|u|p−2 u , |x − ai |p
k
i=1 μi
891
< μ, ¯ the operator L
1,p
u ∈ W0 (Ω)
is positive and the first eigenvalue λ1 of L is well defined, namely, k |u|p p i=1 μi |x−ai |p ) dx Ω (|∇u| − λ1 = λ1 (Ω) = . inf p 1,p u∈W (Ω)\{0} Ω |u| dx 0
Another challenging problem related to (1.1) is to investigate the extremal functions by which the best constant Sμi is achieved. Here we recall the recent work of Abdellaoui, Felli and Peral [1], where the authors studied the following limiting problem: ⎧ p−1 ⎪ ∗ ⎨ − u − μ u = up −1 in RN \ {ai }, p i p |x − ai | ⎪ ⎩ u ∈ D 1,p RN , u > 0 in RN \ {ai }. They proved that for 0 μi < μ¯ and 1 < p < N , the problem has radially symmetric ground states
p−N p−N x − ai |x − ai | ai p p =ε , ∀ε > 0, (1.4) Vp,μi ,ε (x) = ε Up,μi Up,μi ε ε that satisfy
∇V ai RN
ai p p − μi |Vp,μi ,ε (x)| (x) p,μi ,ε |x − ai |p
=
a V i
p ∗ N = (Sμ ) p . i
p,μi ,ε (x)
(1.5)
RN
Moreover, the positive radial functions Up,μi have the following properties: lim r a(μi ) Up,μi (r) = C1 > 0,
(1.6)
lim r b(μi ) Up,μi (r) = C2 > 0, (r) = C1 a(μi ) 0, lim r a(μi )+1 Up,μ i r→0 (r) = C2 b(μi ) > 0, lim r b(μi )+1 Up,μ i
(1.7)
r→0
r→+∞
(1.8) (1.9)
r→+∞
where C1 and C2 are positive constants depending on p and N and a(μi ) and b(μi ) are zeroes of the function f (t) = (p − 1)t p − (N − p)t p−1 + μi ,
t 0,
(1.10)
that satisfy 0 a(μi ) <
N −p < b(μi ), p
i = 1, 2, . . . , k.
The above results are crucial to the study of problem (1.1). For example, in the recent work of Han [13], a particular case of problem (1.1) was studied, the case when k = 1 and q = p. The
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author applied properties (1.4)–(1.10) and obtained the existence of positive solutions by variational arguments. We mention that the singular quasilinear problems related to Hardy inequality were investigated not only in [1] and [13] but also in [11] and [12] and in the references therein. Now we recall another important singular problem, to which much attention has been paid in the past several years, namely, u ∗ −u − μ 2 = |u|2 −2 u + λ|u|q−2 u, x ∈ Ω, |x| (1.11) u = 0, x ∈ ∂Ω, where Ω is a smooth bounded domain in RN (N 3), 0 ∈ Ω, λ > 0, 0 μ < (N − 2)2 /4, 2 q < 2∗ and 2∗ = 2N/(N − 2) is the critical Sobolev exponent. This problem was studied in [4,6–8,10,15–17,20,22] and in the references therein. Many important results on the nontrivial solutions of (1.11) were obtained in these publications and these results give us very good insight into the problem. Most recently, the authors of [5] and [9] studied the following semilinear problem with multiple Hardy-type terms: ⎧ k ⎪ μi ∗ ⎨ u = |u|2 −2 u, x ∈ Ω, −u − 2 (1.12) |x − ai | i=1 ⎪ ⎩ u = 0, x ∈ ∂Ω, where Ω is also a bounded domain in RN with smooth boundary, N 3, ai ∈ Ω and 0 μi < (N − 2)2 /4 for i = 1, 2, . . . , k and k 2. By applying the Hardy inequality and critical point theory, some elegant results about problem (1.12) were obtained, such as the existence and multiplicity of solutions, the asymptotic behaviors of solutions and so on. These results enable us to understand problem (1.11) better. Therefore, an interesting question arises naturally, that is, whether the results about the semilinear equation (1.12) remain true for (1.1), the quasilinear problem with a critical Sobolev exponent and multiple Hardy-type terms? Stimulated by [1,5,9,13], in this paper we shall study the existence of solutions of problem (1.1) and give some positive answers to the above question. To our knowledge, there are no results on the existence of nontrivial solutions for (1.1) for k 2. It is therefore meaningful for us to investigate problem (1.1) deeply. However, due to the singularities caused by the terms |x − ai |−p (i = 1, 2, . . . , k), the quasilinear problem (1.1) becomes more complicated to deal with than (1.12) and therefore we have to face more difficulties. Now we recall the following standard definition. Assume that X is a Banach space, X −1 is the dual space of X. The functional I ∈ C 1 (X, R) is said to satisfy the Palais–Smale condition at level c ((PS)c in short), if every sequence {un } ⊂ X satisfying I (un ) → c and I (un ) → 0 in X −1 has a convergent subsequence. 1,p In this paper, we will take I = J and X = W0 (Ω). To proceed, we need the following assumptions: (H1 ) There exists an l ∈ {1, 2, . . . , k} such that (Sμl )
N p
N p−N p−N p Q(al ) p = min (Sμi ) Q(ai ) p , i = 1, 2, . . . , k .
(H2 ) Q(x) is a positive bounded function on Ω and there exists an x0 ∈ Ω such that Q(x0 ) is a strict local maximum. Furthermore,
Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
893
Q(x0 ) = QM = max Q(x), Ω Q(x) − Q(x0 ) = o |x − x0 |p as x → x0 and
Q(x) − Q(al ) = o |x − al |p as x → al . ¯ (H3 ) 0 < μi < μ¯ for every i = 1, 2, . . . , k and ki=1 μi < μ. Now we are ready to state the main results of this paper, which are concluded as the following theorem. By the fact that b(μl ) > (N − p)p −1 we can verify that the intervals for q and μl in the following theorem are allowable. To the best of our knowledge, the results are new in the case when k 2 and p = 2. Theorem 1.1. Suppose (H1 ), (H2 ) and (H3 ). Assume that one of the following conditions holds: (i) λ > 0 and q¯ < q < p ∗ , where p(2N − b(μl )p − p) N , . q¯ = max p, b(μl ) N −p (ii) N > p 2 , q = p, 0 < λ < λ1 and 0 < μl (N − p 2 )N p−1 p −p . (iii) N > p 2 , p < q < p ∗ , λ 0 and 0 < μl (N − p 2 )N p−1 p −p . 1,p
Then problem (1.1) has at least one positive solution u ∈ W0 (Ω). The proof of Theorem 1.1 will be given in Section 2. At the end of this section, we explain 1,p 1,p some notations employed in this paper: (W0 (Ω))−1 is the dual space of W0 (Ω), O(ε t ) denotes the quantity satisfying |O(ε t )|/ε t C and o(ε t ) means |o(ε t )|/ε t → 0 as ε → 0. In the following argument, we employ C to denote the positive constant and omit dx in integral for convenience. 2. Proof of the main results In this section, we will prove Theorem 1.1 by employing the mountain pass theorem due to Ambrosetti and Rabinowitz [2]. For this purpose, we need to establish several preliminary lemmas. Lemma 2.1. The functional J satisfies (PS)c condition with N N p−N p−N 1 p p c < c∗ = min (Sμl ) Q(al ) p , (S0 ) (QM ) p . N Proof. The argument is similar to that of Lemma 2.1 in [13] and a suitable version of the concentration compactness principle [18,19] will be employed, here we omit the details. 2 In the following, we investigate the extremal function defined in (1.4). To this end, we define the function
p−N x − ai ai ai p , uμi ,ε (x) = ϕai (x)Vp,μi ,ε (x) = ε ϕai (x)Up,μi ε
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where the cut-off function ϕai ∈ C0∞ (Bρ (ai )), 0 ϕai 1 and ϕai ≡ 1 for all x ∈ Bρ/2 (ai ). The radius ρ is chosen small enough such that 0 < ρ < min |ai − aj |, i, j = 1, 2, . . . , k, i = j and Bρ (ai ) ⊂ Ω. Then there exist positive constants C1 and C2 such that C1 |x − aj | C2 ,
∀x ∈ Bρ (ai ), 1 j k, j = i.
ai If ε → 0, the behavior of uaμii ,ε (x) has to be the same as that of the extremal function Vp,μ i ,ε (x), but we need the precise estimates of the error terms. To this end, setting
βi = b(μi ) −
N −p , p
i = 1, 2, . . . , k,
then we have the following estimates (2.1)–(2.4). The proof of Lemma 2.2 follows the same line as that of [13], here we omit it. Lemma 2.2. For ε > 0 small and any i, j = 1, 2, . . . , k with i = j, we have the following estimates:
ai p a p N ∇u i − μi |uμi ,ε | = (Sμi ) p + O ε pβi , (2.1) μi ,ε |x − ai |p Ω a p ∗ N u i = (Sμ ) p + O ε p∗ βi , (2.2) μi ,ε i Ω
⎧ N +(1− N N ∗ ⎪ p )q , Cε ⎪ ⎪ b(μi ) < q < p , ⎨ a q u i N +(1− N N p )q |ln ε|, Cε q = b(μ , μi ,ε ⎪ i) ⎪ ⎪ Ω ⎩ Cε qβi , N p q < b(μ , i) ⎧ Cε p , b(μi ) > Np , ⎪ ⎪ ⎨ |uaμii ,ε |p Cε p |ln ε|, b(μi ) = Np , |x − aj |p ⎪ ⎪ ⎩ Cε pβi , Ω b(μ ) < N . i
(2.3)
(2.4)
p
Furthermore, as ε → 0 we have a q u i → 0, p q < p ∗ μi ,ε Ω
and
Ω
|uaμii ,ε |p → 0, |x − aj |p
i, j = 1, 2, . . . , k with i = j.
For further discussion, we recall the following result of Talenti [21]: For all ε > 0, a ∈ RN and N−p CN,p = N(N − p)(p − 1)−1 p2 ,
Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
895
the best Sobolev constant S0 is achieved by the extremal functions p−N p p−N a = CN,p ε p 1 + |x − a| p−1 p , Vp,0,ε that satisfy ∗ N ∇V a p = V a p = (S0 ) p . p,0,ε p,0,ε RN
RN
a We deal with the function Vp,0,ε by the method similar to that of Lemma 2.2. To this end, for all a ∈ Ω \ {a1 , a2 , . . . , ak } we define the function a ua0,ε (x) = ϕ(x)Vp,0,ε (x),
where the cut-off function ϕ(x) ∈ C0∞ (Bρ (a)), 0 ϕ(x) 1 and ϕ(x) ≡ 1 for all x ∈ Bρ/2 (a). The radius ρ is chosen small enough such that 0 < ρ < min |a − ai |, i = 1, 2, . . . , k and Bρ (a) ⊂ Ω. Then there exist positive constants C1 and C2 such that C1 |x − ai | C2 ,
∀x ∈ Bρ (a), i = 1, 2, . . . , k.
Furthermore, by taking μi = 0 in (2.1)–(2.4) and from the fact that b(0) = (N − p)/(p − 1) we have the following lemma. Lemma 2.3. For ε > 0 small and every i ∈ {1, 2, . . . , k} we have the following estimates: a p N N−p ∇u = (S0 ) p + O ε p−1 , (2.5) 0,ε Ω
a p ∗ N N u = (S0 ) p + O ε p−1 , 0,ε
(2.6)
Ω
⎧ N +(1− N )q N (p−1) ∗ p ⎪ Cε , ⎪ N −p < q < p , ⎪ ⎨ a q N u Cε N +(1− p )q |ln ε|, q = N (p−1) , 0,ε N −p ⎪ ⎪ ⎪ Ω ⎩ q(N−p) Cε p(p−1) , p q < NN(p−1) −p , ⎧ p 2 N >p , ⎪ ⎨ Cε , |ua0,ε |p p Cε |ln ε|, N = p 2 , |x − ai |p ⎪ ⎩ N−p Ω Cε p−1 , N < p2 . Furthermore, as ε → 0 we have a q u → 0, p q < p ∗ 0,ε Ω
and
Ω
|ua0,ε |p |x − ai |p
→ 0.
(2.7)
(2.8)
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Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
Now we are in the position to verify the properties of the functional J . The argument is based on the analytical techniques and the results of Lemmas 2.2 and 2.3. Lemma 2.4. Suppose (H1 ), (H2 ) and (H3 ). Assume that one of the following conditions holds: (i) λ > 0 and q¯ < q < p ∗ , where p(2N − b(μl )p − p) N , q¯ = max p, . b(μl ) N −p (ii) N > p 2 , q = p, 0 < λ < λ1 and 0 < μl (N − p 2 )N p−1 p −p . (iii) N > p 2 , p < q < p ∗ , λ 0 and 0 < μl (N − p 2 )N p−1 p −p . 1,p
Then there exists a nonnegative function v ∈ W0 (Ω) \ {0} such that sup J (tv) < c∗ , t0
where c∗ is defined as in Lemma 2.1. Proof. To prove the lemma, we need to distinguish the following cases: N p
Case I. (Sμl ) (Q(al )) and N p
p−N p
N p
< (S0 ) (QM ) N p
p−N
p−N p
p−N
Case II. (Sμl ) (Q(al )) p (S0 ) (QM ) p . We first study Case I. The definition of c∗ implies that c∗ =
N p−N 1 p (Sμl ) Q(al ) p . N
Then the function [0, +∞): tp g(t) ¯ = p Ω
a
uμll ,ε in Lemma 2.2 is used to define the following functions on the interval
∗ al p p ∗ a p tp ∇u l − μl |uμl ,ε | − Q(x)uaμll ,ε μl ,ε p ∗ |x − al | p Ω
and g(t) = J tuaμll ,ε = g(t) ¯ −
k |uaμll ,ε |p tp t q al q uμl ,ε . μi − λ p |x − ai |p q i =l,i=1
Ω
Ω
For ρ > 0 small and η > 0, it follows from (H2 ) that ∗ ∗ ∗ Q(x)ual p − Q(al )ual p Q(x) − Q(al )ual p μl ,ε μl ,ε μl ,ε Ω
Ω
Ω
Cη
p ∗ |x − al |p uaμll ,ε
Ω
Cηε p .
Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
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By (2.2) we have p∗ N ∗ Q(x)uaμll ,ε = Q(al )(Sμl ) p + O ε p βl + o ε p . Ω
Then from the fact that
p ∗ t 1 N p−N tp sup B1 − ∗ B2 = B1p B2 p , p N t0 p
B1 > 0, B2 > 0
we obtain 1 sup g(t) ¯ = N t0
N p−N
al p p a p ∗ p a p l ∇u l − μl |uμl ,ε | Q(x) uμl ,ε μl ,ε |x − al |p
Ω
Ω
N N N p−N ∗ 1 (Sμl ) p + O ε pβl p Q(al )(Sμl ) p + O ε p βl + o ε p p = N N p−N 1 (Sμl ) p Q(al ) p + O ε pβl + o ε p . N
Furthermore, the fact that limt→+∞ g(t) = −∞ and g(t) > 0 as t → 0+ implies supt0 g(t) must be attained at some finite tε > 0. Then from g (tε ) = 0 we deduce that
al p a p ∇u l − μl |uμl ,ε | μl ,ε |x − al |p Ω
= tεp
∗ −p
p∗ Q(x)uaμll ,ε + λtεq−p
Ω
k a q u l + μ i μl ,ε i =l,i=1
Ω
Ω
|uaμll ,ε |p . |x − ai |p
Hence, we obtain from Lemma 2.2 that (2.9)
C1 < tε < C2 , where C1 and C2 are positive constants independent of ε. To continue, we consider the following cases. (i) λ > 0 and q¯ < q < p ∗ , where p(2N − b(μl )p − p) N , . q¯ = max p, b(μl ) N −p In this case, from Lemma 2.2 we have a |uμll ,ε |p lim = 0, i = 1, 2, . . . , k, i = l. ε→0 |x − ai |p Ω
Consequently, p
g(tε ) = g(t ¯ ε) −
tε p
k i =l,i=1
μi Ω
q
|uaμll ,ε |p tε −λ p |x − ai | q
a q u l μl ,ε
Ω
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Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903 q
sup g(t) ¯ −λ t0
tε q
a q u l μl ,ε
Ω
N p−N 1 (Sμl ) p Q(al ) p + O ε pβl + o ε p − C N
a q u l . μl ,ε
Ω
From (2.3) and the assumption q¯ < q a q N u l Cε N +(1− p )q μl ,ε
< p∗
we obtain
Ω
and
N q < min{p, pβl }. N + 1− p
Taking ε small enough we have g(tε ) <
N p−N 1 (Sμl ) p Q(al ) p . N
(ii) q = p, 0 < λ < λ1 and b(μ)
(2.10)
N p.
The same argument as that in case (i) shows that λ p al p uμl ,ε ¯ ε ) − tε g(tε ) g(t p Ω N p−N 1 (Sμl ) p Q(al ) p + O ε pβl + o ε p − C N
Ω
If b(μl ) > Np , from (2.3) we obtain a p u l Cε p . μl ,ε Ω
Then (2.10) follows by taking ε small enough. If b(μl ) = Np , from the fact a p u l Cε p |ln ε| μl ,ε Ω
we obtain that (2.10) holds for ε small. (iii) N > p 2 , λ 0 and p < q < p ∗ . In this case, from Lemma 2.2 we obtain p
¯ ε) − g(tε ) g(t
tε p
k i =l,i=1
μi Ω
|uaμll ,ε |p |x − ai |p
a p u l . μl ,ε
Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
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k N p−N |uaμll ,ε |p 1 . (Sμl ) p Q(al ) p + O ε pβl + o ε p − C N |x − ai |p
i =l,i=1 Ω
If b(μl ) >
N p,
k i =l,i=1 Ω
(2.4) implies that
|uaμll ,ε |p Cε p . |x − ai |p
Hence, (2.10) holds naturally for ε small. If b(μl ) = Np , from (2.4) we obtain k i =l,i=1 Ω
|uaμll ,ε |p Cε p |ln ε|. |x − ai |p
Thus (2.10) follows by taking ε small enough. On the other hand, it is easy to verify that the function f (t) = (p − 1)t p − (N − p)t p−1 + μl ,
t ∈ [0, +∞)
has the only minimal point t¯ = (N − p)/p and is increasing on the interval (t¯, +∞). Thus for N > p 2 we have
N N N p−1 (N − p 2 ) b(μl ) ⇐⇒ f f b(μl ) = 0 ⇐⇒ 0 < μl . p p pp Hence Case I is verified. Next, we investigate Case II. In this case we have p−N N N N p−N p−N 1 1 1 c∗ = (S0 ) p (QM ) p = (S0 ) p Q(x0 ) p (Sμl ) p Q(al ) p , N N N where x0 is the maximum point of Q(x) defined as in (H2 ). If x0 = ai for some i ∈ {1, 2, . . . , k}, from the fact that Sμi < S0 we obtain N N N p−N p−N p−N 1 1 1 (S0 ) p Q(ai ) p > (Sμi ) p Q(ai ) p (Sμl ) p Q(al ) p , N N N which is impossible. Hence x0 = ai for any i ∈ {1, 2, . . . , k}. x0 In the following argument, we employ the function u0,ε given in Lemma 2.3. Consider the functions defined on the interval [0, +∞): p p∗ x0 p∗ ∇ux0 p − t ¯h(t) = t Q(x)u0,ε 0,ε p p∗
c∗ =
Ω
Ω
and x0 p k p |u0,ε | x0 t q x0 q ¯ −t u0,ε . h(t) = J tu0,ε = h(t) μi − λ p |x − ai |p q i=1
Then from (2.5)–(2.8) and (H2 ) we have
Ω
Ω
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Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
¯ = 1 sup h(t) N t0
x0 p ∇u
N p
0,ε
Ω
x0 p∗ Q(x)u0,ε
− N−p p
Ω
N N − N−p N−p N N 1 p (S0 ) p + O ε p−1 p Q(x0 )(S0 ) p + O ε p−1 + o ε p = N N p−N N−p 1 (S0 ) p Q(x0 ) p + O ε p−1 + o ε p . N Arguing as in Case I, we have that supt0 h(t) must be attained at some tε and there exist positive constants C1 and C2 such that C1 < tε < C2 . Furthermore,
¯ − h(tε ) sup h(t) t0
x0 p p k q |u0,ε | tε x0 q tε u0,ε μi − λ p |x − ai |p q i=1
Ω
Ω
N p−N N−p 1 (S0 ) p Q(x0 ) p + O ε p−1 + o ε p N x0 p k |u0,ε | x0 q u . μi − C −C 0,ε |x − ai |p
i=1
Ω
(2.11)
Ω
To proceed, we need to investigate the following cases. (i) λ > 0 and q¯0 < q < p ∗ , where N (p − 1) ∗ p ,p − . q¯0 = max p, N −p p−1 From Lemma 2.3 we have x0 p |u0,ε | lim = 0, i = 1, 2, . . . , k. ε→0 |x − ai |p Ω
Then (2.11) implies N p−N N−p 1 h(tε ) (S0 ) p Q(x0 ) p + O ε p−1 + o ε p − C N
x0 q u . 0,ε
Ω
Furthermore, from (2.7) and the assumption q¯0 < q a q N u l Cε N +(1− p )q μl ,ε
< p∗
we deduce that
Ω
and
N N −p N + 1− q < min p, . p p−1
Consequently, N p−N 1 (S0 ) p Q(x0 ) p N for ε small enough.
h(tε ) <
(2.12)
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901
On the other hand, from the equality (p − 1)b(μl )p − (N − p)b(μl )p−1 + μl = 0 we obtain
(p − 1)b(μl )
p−1
N −p b(μl ) − p−1
+ μl = 0.
Then the following inequalities N −p N −p < b(μl ) < p p−1 and N p(2N − b(μl )p − p) p N(p − 1) ∗ ,p − max p, , max p, N −p p−1 b(μl ) N −p hold for all μl ∈ (0, μ). ¯ Furthermore, q¯0 q¯
and (q, ¯ p ∗ ) ⊂ (q¯0 , p ∗ ),
where the constant q¯ is defined as in Case I. Hence q ∈ (q¯0 , p ∗ ) under the stronger hypothesis q ∈ (q, ¯ p ∗ ) of Case I. (ii) N > p 2 , q = p and 0 < λ < λ1 . N −p p−1 .
Then (2.7) and (2.11) imply that x0 p N p−N N−p 1 h(tε ) (S0 ) p Q(x0 ) p + O ε p−1 + o ε p − C u0,ε N
In this case we have p <
Ω
N p−N N−p 1 (S0 ) p Q(x0 ) p + O ε p−1 + o ε p − Cε p . N Thus (2.12) holds naturally for ε small.
(iii) N > p 2 , λ 0 and p < q < p ∗ . From (2.8) and (2.11) and by the fact that p < h(tε )
N −p p−1
we obtain
k N p−N N−p 1 (S0 ) p Q(x0 ) p + O ε p−1 + o ε p − C N
i=1 Ω
x
0 p |u0,ε |
|x − ai |p
N p−N N−p 1 (S0 ) p Q(x0 ) p + O ε p−1 + o ε p − Cε p . N Then (2.12) follows by taking ε small enough. Hence, Case II is proved. From Cases I and II we conclude Lemma 2.4. 2
Proof of Theorem 1.1. We verify that the functional J satisfies the mountain pass geometry. To this end, we consider the energy level c = inf max J γ (t) , γ ∈Γ t∈[0,1]
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Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
where 1,p Γ = γ ∈ C [0, 1], W0 (Ω) γ (0) = 0, J γ (1) < 0 . If λ 0 and p < q < p ∗ , Hardy inequality and Sobolev inequality imply that
k μi |u|p λ 1 p p∗ |u|q |∇u| − − ∗ Q(x)|u| − |x − ai |p p q
1 J (u) = p
i=1
Ω
1 1 μi 1− p μ¯ k
i=1
−
λ −q (S0,q ) p q
Ω
Ω
∗ QM −p |∇u| − ∗ (S0 ) p p
Ω
p
p
Ω
q
|∇u|p
p∗ |∇u|
p
p
.
Ω
If 0 < λ < λ1 and q = p, then by the definition of λ1 we have that 1 J (u) = p
k μi |u|p λ 1 p p∗ Q(x)|u| − |u|p |∇u| − − |x − ai |p p∗ p i=1
Ω
Ω
Ω
p∗ k p p∗ λ 1 Q 1 M − 1− μi |∇u|p − ∗ (S0 ) p |∇u|p . 1− p λ1 μ¯ p
i=1
Ω
Ω
Hence, there exists ρ¯ > 0 small enough such that b = inf J (u) > 0 = J (0). u=ρ¯
1,p
Now we employ the function v ∈ W0 (Ω) \ {0} obtained in Lemma 2.4. Then there exists t0 > 0 such that t0 v > ρ¯ and J (t0 v) < 0. By the mountain pass theorem [2], there exists a 1,p sequence {un } ⊂ W0 (Ω) such that J (un ) → 0
J (un ) → c,
1,p −1 in W0 (Ω) .
From Lemma 2.4 we obtain c sup J (tt0 v) sup J (tv) < c∗ . t∈[0,1]
t0
According to Lemma 2.1, there exists a subsequence of {un } (still denoted by {un }), such that 1,p un → u strongly in W0 (Ω). Then u is a critical point of the functional J and thereby a solution of problem (1.1), c is the corresponding critical value of J . In order to find the positive solution of (1.1), we replace the functional J (u) with the following J + (u):
k + p∗ λ + q μi |u|p 1 1 + p u Q(x) u − , J (u) = |∇u| − − p |x − ai |p p∗ q Ω
i=1
Ω
Ω
Y. Cao, D. Kang / J. Math. Anal. Appl. 333 (2007) 889–903
903
where u+ = max{u, 0} ∈ W0 (Ω). Repeating the above argument, we obtain a critical point u of J + , which satisfies the following equation ⎧ k ⎪ p∗ −1 q−1 |u|p−2 u ⎨ −p u − μi = Q(x) u+ + λ u+ , x ∈ Ω, p |x − ai | ⎪ i=1 ⎩ u = 0, x ∈ ∂Ω. 1,p
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