Quasilinear Elliptic Equations with Hardy-Sobolev Critical Exponents

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 482740, 6 pages http://dx.doi.org/10.1155/2014/482740

Research Article Quasilinear Elliptic Equations with Hardy-Sobolev Critical Exponents: Existence and Multiplicity of Nontrivial Solutions Guanwei Chen School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, China Correspondence should be addressed to Guanwei Chen; [email protected] Received 30 November 2013; Accepted 12 January 2014; Published 19 February 2014 Academic Editor: Tai-Ping Chang Copyright © 2014 Guanwei Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the existence of positive solutions and multiplicity of nontrivial solutions for a class of quasilinear elliptic equations by using variational methods. Our obtained results extend some existing ones.

In the case where 𝑠 = 0 and 𝜇 = 1 hold, then (1) reduces to the quasilinear elliptic problem:

1. Introduction and Main Results Let us consider the following problem:

−Δ𝑝 𝑢 − |𝑢|𝑝

∗

|𝑢|𝑝 (𝑠)−2 −Δ 𝑝 𝑢 − 𝜇 𝑢 = 𝑓 (𝑥, 𝑢) , |𝑥|𝑠 𝑢 = 0,

𝑥 ∈ Ω \ {0} ,

(1)

𝑥 ∈ 𝜕Ω,

where Δ 𝑝 𝑢 = div(|∇𝑢|𝑝−2 ∇𝑢) denotes the 𝑝-Laplacian differential operator, Ω is an open bounded domain in R𝑁 (𝑁 ≥ 3) with smooth boundary 𝜕Ω and 0 ∈ Ω, 0 ≤ 𝑠 < 𝑝, 𝑝∗ (𝑠) = 𝑝(𝑁 − 𝑠)/(𝑁 − 𝑝) is the Hardy-Sobolev critical exponent, and 𝑝∗ = 𝑝∗ (0) = 𝑁𝑝/(𝑁 − 𝑝) is the Sobolev critical exponent. Here, we let ‖𝑢‖ := (∫ |∇𝑢|𝑝 𝑑𝑥)

1/𝑝

Ω

,

(2)

which is equivalent to the usual norm of Sobolev space 1,𝑝 𝑊0 (Ω) due to the Poincar´e inequality. Let 𝐴 𝑠 (Ω) :=

inf 1,𝑝

‖𝑢‖𝑝 𝑝∗ (𝑠)

𝑢∈𝑊0 (Ω)\{0} (∫ (|𝑢| Ω

/|𝑥|𝑠 )𝑑𝑥)

which is the best Hardy-Sobolev constant.

𝑝/𝑝∗ (𝑠)

,

(3)

∗

−2

𝑢 = 𝑓 (𝑥, 𝑢) ,

𝑢 = 0,

𝑥 ∈ 𝜕Ω.

𝑥 ∈ Ω,

(4)

Gonc¸alves and Alves [1] have studied (4) in R𝑁 involving 𝑓(𝑥, 𝑢) = ℎ(𝑥)𝑢𝑞 , 𝑢 ≥ 0 and 𝑢 ≢ 0 to obtain existence of positive solutions where 2 ≤ 𝑝 < 𝑁, 0 < 𝑞 < 𝑝 − 1, or 𝑝 − 1 < 𝑞 < 𝑝∗ − 1 and a suitable ℎ. We should mention that problem (4) with 𝑝 = 2 has been widely studied since Br´ezis and Nirenberg; see [2–4] and the references therein. Ghoussoub and Yuan [5] have studied (1) with 𝑓(𝑥, 𝑢) = 𝜆|𝑢|𝑟−2 𝑢, where 𝑝 < 𝑟 < 𝑝∗ . They obtained a positive solution in the case where 𝜆 > 0, 𝜇 > 0, and 𝑁 > [𝑝(𝑝 − 1)𝑟 + 𝑝2 ]/[𝑝 + (𝑝 − 1)(𝑟 − 𝑝)] (in particular if 𝑁 ≥ 𝑝2 ) hold. They also obtained a sign-changing solution in the case where 𝜆 > 0, 𝜇 > 0, and 𝑁 > [𝑝(𝑝 − 1)𝑟 + 𝑝]/[1 + (𝑝 − 1)(𝑟 − 𝑝)] (in particular if 𝑁 > 𝑝3 − 𝑝2 + 𝑝) hold. For other relevant papers, see [6–12] and the references therein. We should mention that the energy functional associated 1,𝑝 with (1) is defined on 𝑊0 (Ω), which is not a Hilbert space for 𝑝 ≠ 2. Due to the lack of compactness of the embedding ∗ ∗ 1,𝑝 1,𝑝 in 𝑊0 (Ω) 󳨅→ 𝐿𝑝 (Ω) and 𝑊0 (Ω) 󳨅→ 𝐿𝑝 (𝑠) (Ω, |𝑥|−𝑠 𝑑𝑥), we cannot use the standard variational argument directly. The corresponding energy functional fails to satisfy the classical 1,𝑝 Palais-Smale ((PS) for short) condition in 𝑊0 (Ω). However,

2

Journal of Applied Mathematics

a local (PS) condition can be established in a suitable range. Then the existence result is obtained via constructing a minimax level within this range and the Mountain Pass Lemma due to Ambrosetti et al. [2] and Rabinowitz [13]. In this paper, we study (1) with a general nonlinearity by using a variational method; besides, we also considerably generalize the results obtained in [5]. In what follows, we always assume that the nonlinearity 𝑓 satisfies 𝑓(𝑥, 0) ≡ 0. 𝑡 Let 𝐹(𝑥, 𝑡) := ∫0 𝑓(𝑥, 𝑠)𝑑𝑠, 𝑥 ∈ Ω. To state our main results, we still need the following assumptions. (𝐴 1 ) 𝑓 ∈ 𝐶(Ω × R+ , R), lim𝑡 → 0+ (𝑓(𝑥, 𝑡)/𝑡𝑝−1 ) = 0, and ∗ lim𝑡 → ∞ (𝑓(𝑥, 𝑡)/𝑡𝑝 −1 ) = 0 uniformly for 𝑥 ∈ Ω. (𝐴 2 ) There exists a constant 𝜌 with 𝜌 > 𝑝 such that 0 < 𝜌𝐹 (𝑥, 𝑡) ≤ 𝑓 (𝑥, 𝑡) 𝑡,

∀𝑥 ∈ Ω, ∀𝑡 ∈ R+ \ {0} .

The rest of this paper is organized as follows. In Section 2, we give some preliminary lemmas, which are useful in the proofs of our main results. In Section 3, we give the detailed proofs of our main results.

2. Preliminaries In what follows, we let ‖ ⋅ ‖𝑝 denote the norm in 𝐿𝑝 (Ω). It is obvious that the values of 𝑓(𝑥, 𝑡) for 𝑡 < 0 are irrelevant in Theorem 1, so we may define 𝑓 (𝑥, 𝑡) ≡ 0

∀𝑥 ∈ Ω, ∀𝑡 ∈ R \ {0} .

(5)

(6)

𝑝∗ (𝑠)−1

(𝑢+ ) −Δ 𝑝 𝑢 = 𝜇 |𝑥|𝑠

(10)

𝑥 ∈ 𝜕Ω.

𝑝∗ (𝑠)

𝑝 1 ) , 𝑝∗ − } 𝑝 𝑝−1

(𝑢+ ) 𝜇 1 𝐼 (𝑢) = ∫ |∇𝑢|𝑝 𝑑𝑥 − ∗ ∫ 𝑝 Ω 𝑝 (𝑠) Ω |𝑥|𝑠 − ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥, Ω

(7)

hold, then (1) has at least one positive solution. Theorem 2. Suppose that 𝑁 ≥ 3, 0 < 𝜇 < ∞, 0 ≤ 𝑠 < 𝑝, and 1 < 𝑝 < 𝑁 hold. If (𝐴 3 ), (𝐴 4 ), and (7) hold, then (1) has at least two distinct nontrivial solutions. Noting that 𝜌 > 𝑝 and 𝑝2 ≤ 𝑁 imply that (7) holds, therefore, we have the following corollaries. Corollary 3. Suppose that 0 < 𝜇 < ∞, 0 ≤ 𝑠 < 𝑝, 𝑝2 ≤ 𝑁, and 1 < 𝑝 < 𝑁. Moreover, (𝐴 1 ) and (𝐴 2 ) hold; then (1) has at least one positive solution. Corollary 4. Suppose that 0 < 𝜇 < ∞, 0 ≤ 𝑠 < 𝑝, 𝑝 ≤ 𝑁, and 1 < 𝑝 < 𝑁. Moreover, (𝐴 3 ) and (𝐴 4 ) hold; then (1) has at least two distinct nontrivial solutions. Remark 5. Theorem 1 generalizes Theorem 1.3 in [5], where the author only studied the special situation that 𝑓(𝑥, 𝑢) = 𝜆|𝑢|𝑟−2 𝑢, 𝑝 < 𝑟 < 𝑝∗ . There are functions 𝑓 satisfying the assumptions of our Theorem 1 and not satisfying those in [5]. Let (𝑥, 𝑡) ∈ Ω × R,

(8)

𝑑𝑥 (11)

1,𝑝

𝑢 ∈ 𝑊0 (Ω) .

By Hardy-Sobolev inequalities (see [5, 14]) and (𝐴 1 ), we know 1,𝑝 𝐼 ∈ 𝐶1 (𝑊0 (Ω), R). Now it is well known that there exists a one-to-one correspondence between the weak solutions of 1,𝑝 (10) and the critical points of 𝐼 on 𝑊0 (Ω). More precisely 1,𝑝 we say that 𝑢 ∈ 𝑊0 (Ω) is a weak solution of (10), if, for any 1,𝑝 V ∈ 𝑊0 (Ω), there holds 󸀠

𝑝−2

⟨𝐼 (𝑢) , V⟩ = ∫ |∇𝑢| Ω

𝑝∗ (𝑠)−1

(𝑢+ ) (∇𝑢, ∇V) 𝑑𝑥 − 𝜇 ∫ |𝑥|𝑠 Ω

V𝑑𝑥

− ∫ 𝑓 (𝑥, 𝑢) V𝑑𝑥 = 0. Ω

2

∞

𝑥 ∈ Ω \ {0} ,

The energy functional corresponding to (10) is given by

Theorem 1. Suppose that 𝑁 ≥ 3, 0 < 𝜇 < ∞, 0 ≤ 𝑠 < 𝑝, and 1 < 𝑝 < 𝑁 hold. If (𝐴 1 ), (𝐴 2 ), and

𝑓 (𝑥, 𝑡) := 𝑔 (𝑥) |𝑡|𝑘−2 𝑡 + 𝑎|𝑡|𝑙−2 𝑡,

+ 𝑓 (𝑥, 𝑢) ,

𝑢 = 0,

Now, our main results read as follows.

𝜌 > max {𝑝, 𝑝∗ (1 −

(9)

We firstly consider the existence of nontrivial solutions to the problem:

(𝐴 3 ) 𝑓 ∈ 𝐶(Ω × R, R), lim𝑡 → 0 (𝑓(𝑥, 𝑡)/|𝑡|𝑝−1 ) = 0, and ∗ lim|𝑡| → ∞ (𝑓(𝑥, 𝑡)/|𝑡|𝑝 −1 ) = 0 uniformly for 𝑥 ∈ Ω. (𝐴 4 ) There exists a constant 𝜌 with 𝜌 > 𝑝 such that 0 < 𝜌𝐹 (𝑥, 𝑡) ≤ 𝑓 (𝑥, 𝑡) 𝑡,

for 𝑥 ∈ Ω, 𝑡 ≤ 0.

(12)

Lemma 6 (see [15]). If 𝑓𝑛 → 𝑓 a.e. in Ω and ‖𝑓𝑛 ‖𝑝 ≤ 𝐶 < ∞ for all n and some 0 < 𝑝 < ∞, then 󵄩 󵄩𝑝 󵄩 󵄩𝑝 󵄩 󵄩𝑝 lim (󵄩󵄩󵄩𝑓𝑛 󵄩󵄩󵄩𝑝 − 󵄩󵄩󵄩𝑓𝑛 − 𝑓󵄩󵄩󵄩𝑝 ) = 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑝 .

𝑛→∞

(13)

Lemma 7. For any 𝑎 > 0, 0 ≤ 𝑏 ≤ 1, and 𝜆 ≥ 1, we have (𝑎 + 𝑏)𝜆 ≤ 𝑎𝜆 + 𝜆(𝑎 + 1)𝜆−1 𝑏. Proof. Let

∗

where 𝑔(𝑥) > 0, 𝑔 ∈ 𝐿 (Ω), 𝑎 > 0, and 𝑝 < 𝑘 < 𝑙 < 𝑝 . Obviously, 𝑓 satisfies all the conditions of Theorem 1 in this paper, while it does not satisfy the conditions of Theorem 1.3 in [5].

ℎ (𝑥) := (𝑎 + 𝑥)𝜆 − 𝑎𝜆 − 𝜆(𝑎 + 1)𝜆−1 𝑥. Clearly, ℎ󸀠 (𝑥) < 0, 𝑥 ∈ (0, 1), so ℎ(𝑏) ≤ ℎ(0) = 0.

(14)

Journal of Applied Mathematics

3 𝑝∗

Lemma 8 (see [5]). If 0 ≤ 𝑠 < 𝑝 hold, then we have (i) 𝐴 𝑠 (Ω) is independent of Ω, and will henceforth be denoted by 𝐴 𝑠 ;

By the continuity of embedding, we have ‖𝑢𝑛 ‖𝑝∗ ≤ 𝐶1 < ∞. From [5], going if necessary to a subsequence, one can get that ∇𝑢𝑛 󳨀→ ∇𝑢 a.e. in Ω, 𝑢𝑛 𝑢 ⇀ 𝑥 𝑥

(ii) 𝐴 𝑠 (Ω) is attained when Ω = R𝑁 by the functions (𝑁−𝑝)/𝑝(𝑝−𝑠)

𝑁 − 𝑝 𝑝−1 𝑙𝜀 (𝑥) = (𝜀 (𝑁 − 𝑠) ( ) ) 𝑝−1

(15)

weakly in 𝐿𝑝 (Ω) ,

∗ 󵄨󵄨 󵄨󵄨𝑝∗ (𝑠)−2 𝑢 󵄨𝑢 󵄨 |𝑢|𝑝 (𝑠)−2 𝑢 V𝑑𝑥, ∫ 󵄨 𝑛 󵄨 𝑠 𝑛 V𝑑𝑥 󳨀→ ∫ |𝑥| |𝑥|𝑠 Ω Ω

1,𝑝

∀V ∈ 𝑊0 (Ω) (20)

(𝑝−𝑠)/(𝑝−1) (𝑝−𝑁)/(𝑝−𝑠)

× (𝜀 + |𝑥|

)

for some 𝜀 > 0. Moreover, the functions 𝑙𝜀 (𝑥) are the only positive radial solutions of ∗

𝑢𝑝 (𝑠)−1 −Δ 𝑝 𝑢 = |𝑥|𝑠

(16)

∫

R𝑁

󵄨󵄨 󵄨󵄨𝑝 󵄨󵄨∇𝑙𝜀 󵄨󵄨 𝑑𝑥 = ∫ R𝑁

∗ 1 󵄨 󵄨󵄨 𝜀|𝑡|𝑝 󵄨󵄨𝑓 (𝑥, 𝑡) 𝑡󵄨󵄨󵄨 ≤ 𝑎 (𝜀) + 2𝐶1

for (𝑥, 𝑡) ∈ Ω × (0, +∞) . (21)

Set 𝛿 := 𝜀/2𝑎(𝜀) > 0. When 𝐸 ⊂ Ω, meas(𝐸) < 𝛿, we get 󵄨 󵄨󵄨 ∗ 󵄨󵄨∫ 𝑓 (𝑥, 𝑢 ) 𝑢 𝑑𝑥󵄨󵄨󵄨 ≤ ∫ 𝑎 (𝜀) 𝑑𝑥 + 𝜀 ∫ 󵄨󵄨󵄨𝑢 󵄨󵄨󵄨𝑝 𝑑𝑥 󵄨󵄨 󵄨󵄨 𝑛 𝑛 𝑛󵄨 󵄨 2𝐶1 𝐸 󵄨 󵄨 𝐸 𝐸

in R𝑁, and satisfy 󵄨󵄨 󵄨󵄨𝑝∗ (𝑠) 󵄨󵄨𝑙𝜀 󵄨󵄨 (𝑁−𝑠)/(𝑝−𝑠) 𝑑𝑥 = (𝐴 𝑠 (Ω)) . |𝑥|𝑠

as 𝑛 → ∞. By (𝐴 1 ), we know that for any 𝜀 > 0 there exists 𝑎(𝜀) > 0 such that

(22)

< 𝜀 󳨀→ 0 as meas (𝐸) 󳨀→ 0.

(17)

It follows from Vitali’s theorem that Lemma 9. If (𝐴 1 ), (𝐴 2 ), and 0 < 𝑐 < ((𝑝 − 𝑠)/𝑝(𝑁 − 𝑠))𝐴(𝑁−𝑠)/(𝑝−𝑠) 𝜇(𝑝−𝑁)/(𝑝−𝑠) hold, then 𝐼 satisfies (PS)𝑐 condition. 𝑠 1,𝑝

Proof. Suppose that {𝑢𝑛 } is a (PS)𝑐 sequence in 𝑊0 (Ω). By (𝐴 2 ), we have

Ω

1 󸀠 ⟨𝐼 (𝑢𝑛 ) , 𝑢𝑛 ⟩ 𝜃

as 𝑛 󳨀→ ∞. (23)

Similarly, we can also get

Ω

Ω

as 𝑛 󳨀→ ∞.

(24)

Since 𝐼󸀠 (𝑢𝑛 ) → 0, we have 𝑝∗ (𝑠)

(𝑢𝑛+ ) 1 1 1 󵄩 󵄩𝑝 1 = ( − ) 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩 + ( − ∗ ) 𝜇 ∫ 𝑝 𝜃 𝜃 𝑝 (𝑠) |𝑥|𝑠 Ω

𝑑𝑥

1 − ∫ (𝐹 (𝑥, 𝑢𝑛 ) − 𝑓 (𝑥, 𝑢𝑛 ) 𝑢𝑛 ) 𝑑𝑥 𝜃 Ω ≥(

Ω

∫ 𝐹 (𝑥, 𝑢𝑛 ) 𝑑𝑥 󳨀→ ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥

󵄩 󵄩 𝑐 + 1 + 𝑜 (1) 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩 ≥ 𝐼 (𝑢𝑛 ) −

∫ 𝑓 (𝑥, 𝑢𝑛 ) 𝑢𝑛 𝑑𝑥 󳨀→ ∫ 𝑓 (𝑥, 𝑢) 𝑢 𝑑𝑥

1 1 󵄩󵄩 󵄩󵄩𝑝 − ) 󵄩𝑢 󵄩 , 𝑝 𝜃 󵄩 𝑛󵄩

⟨𝐼󸀠 (𝑢𝑛 ) , 𝑢𝑛 ⟩ 𝑝∗ (𝑠)

(𝑢𝑛+ ) 󵄩 󵄩𝑝 = 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩 − 𝜇 ∫ |𝑥|𝑠 Ω

𝑑𝑥 − ∫ 𝑓 (𝑥, 𝑢𝑛 ) 𝑢𝑛 𝑑𝑥 = 𝑜 (1) . Ω

(25)

Let V𝑛 := 𝑢𝑛 − 𝑢, which together with Lemma 6 implies (18)

∗

where 𝜃 = min{𝜌, 𝑝 (𝑠)}. Hence we conclude that {𝑢𝑛 } is a 1,𝑝 1,𝑝 bounded sequence in 𝑊0 (Ω). So there exists 𝑢 ∈ 𝑊0 (Ω); going if necessary to a subsequence, we have

(1 < 𝛾 < 𝑝∗ ) ,

𝑢𝑛 󳨀→ 𝑢 a.e. in Ω, 𝑛 󳨀→ ∞.

𝑝∗ (𝑠)

(𝑢+ ) 𝑑𝑥 − 𝜇 ∫ |𝑥|𝑠 Ω

𝑑𝑥 (26)

− ∫ 𝑓 (𝑥, 𝑢) 𝑢 𝑑𝑥 = 𝑜 (1) . Ω

From (20), we can obtain

1,𝑝

𝑢𝑛 ⇀ 𝑢 in 𝑊0 (Ω) , 𝑢𝑛 󳨀→ 𝑢 in 𝐿𝛾 (Ω)

𝑝∗ (𝑠)

(V𝑛+ ) 󵄩󵄩 󵄩󵄩𝑝 𝑝 󵄩󵄩V𝑛 󵄩󵄩 + ‖𝑢‖ − 𝜇 ∫ |𝑥|𝑠 Ω

(19)

𝑝∗ (𝑠)

(𝑢+ ) lim ⟨𝐼 (𝑢𝑛 ) , 𝑢⟩ = ‖𝑢‖ − 𝜇 ∫ 𝑛→∞ |𝑥|𝑠 Ω 󸀠

𝑝

𝑑𝑥

− ∫ 𝑓 (𝑥, 𝑢) 𝑢 𝑑𝑥 = 0. Ω

(27)

4

Journal of Applied Mathematics

Note that 𝐼(𝑢𝑛 ) → 𝑐 as 𝑛 → ∞, which together with Lemma 6 implies 𝑝∗ (𝑠)

(V𝑛+ ) 𝜇 1 󵄩 󵄩𝑝 1 𝐼 (𝑢𝑛 ) = 󵄩󵄩󵄩V𝑛 󵄩󵄩󵄩 + ‖𝑢‖𝑝 − ∗ ∫ 𝑝 𝑝 𝑝 (𝑠) Ω |𝑥|𝑠 −

∗ + 𝑝 (𝑠)

(𝑢 ) 𝜇 ∫ (𝑠) Ω |𝑥|𝑠

𝑝∗

∗

so that ∫Ω (|V𝜀 |𝑝 (𝑠) /|𝑥|𝑠 ) = 1. Then, by using the argument as used in [5], we can get the following results: 󵄩 󵄩𝑝 𝐴 𝑠 + 𝐶2 𝜀(𝑁−𝑝)/(𝑝−𝑠) ≤ 󵄩󵄩󵄩V𝜀 󵄩󵄩󵄩 ≤ 𝐴 𝑠 + 𝐶3 𝜀(𝑁−𝑝)/(𝑝−𝑠) ,

𝑑𝑥

𝑑𝑥 − ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥 + 𝑜 (1)

󵄨 󵄨𝑟 𝐶4 𝜀(𝑁−𝑝)𝑟/𝑝(𝑝−𝑠) ≤ ∫ 󵄨󵄨󵄨V𝜀 󵄨󵄨󵄨 𝑑𝑥 ≤ 𝐶5 𝜀(𝑁−𝑝)𝑟/𝑝(𝑝−𝑠) , Ω

Ω

1 ≤ 𝑟 < 𝑝∗ (1 −

∗

𝑝 (𝑠)

= 𝐼 (𝑢) +

(V𝑛+ ) 𝜇 1 󵄩󵄩 󵄩󵄩𝑝 ∫ 󵄩󵄩V𝑛 󵄩󵄩 − ∗ 𝑝 𝑝 (𝑠) Ω |𝑥|𝑠

𝑑𝑥 + 𝑜 (1)

= 𝑐 + 𝑜 (1) .

𝑟 = 𝑝∗ (1 −

Therefore, one gets that

1 ), 𝑝

𝐶4 𝜀((𝑝−1)/(𝑝−𝑠))(𝑁−𝑟(𝑁−𝑝)/𝑝)

𝑝∗ (𝑠)

𝑑𝑥 = 𝑐 + 𝑜 (1) .

(29)

󵄨 󵄨𝑟 ≤ ∫ 󵄨󵄨󵄨V𝜀 󵄨󵄨󵄨 𝑑𝑥 ≤ 𝐶5 𝜀((𝑝−1)/(𝑝−𝑠))(𝑁−𝑟(𝑁−𝑝)/𝑝) , Ω

From (26) and (27), we have

𝑝∗ (1 −

𝑝∗ (𝑠)

(V𝑛+ ) 󵄩󵄩 󵄩󵄩𝑝 󵄩󵄩V𝑛 󵄩󵄩 − 𝜇 ∫ |𝑥|𝑠 Ω

1 ), 𝑝

󵄨 󵄨𝑟 𝐶4 𝜀(𝑁−𝑝)𝑟/𝑝(𝑝−𝑠) |ln 𝜀| ≤ ∫ 󵄨󵄨󵄨V𝜀 󵄨󵄨󵄨 𝑑𝑥 ≤ 𝐶5 𝜀(𝑁−𝑝)𝑟/𝑝(𝑝−𝑠) |ln 𝜀| , Ω

(28) (V𝑛+ ) 𝜇 1 󵄩 󵄩𝑝 𝐼 (𝑢) + 󵄩󵄩󵄩V𝑛 󵄩󵄩󵄩 − ∗ ∫ 𝑝 𝑝 (𝑠) Ω |𝑥|𝑠

(36)

(30)

𝑑𝑥 = 𝑜 (1) ;

1 ) < 𝑟 < 𝑝∗ . 𝑝 (37)

𝑝

then ‖V𝑛 ‖ → 0 as 𝑛 → ∞. Otherwise, there exists a subsequence (still denoted by V𝑛 ) such that 𝑝∗ (𝑠)

(V𝑛+ ) lim 𝜇 ∫ 𝑛→∞ |𝑥|𝑠 Ω

󵄩 󵄩𝑝 lim 󵄩󵄩V 󵄩󵄩 = 𝑘, 𝑛 → ∞󵄩 𝑛 󵄩 By (3), we have 󵄩󵄩 󵄩󵄩𝑝 󵄩󵄩V𝑛 󵄩󵄩 ≥ 𝐴 𝑠 (∫

Ω

|𝑥|

𝑝/𝑝∗ (𝑠)

∀𝑛 ∈ N;

(32)

𝐴(𝑁−𝑠)/(𝑝−𝑠) 𝜇(𝑝−𝑁)/(𝑝−𝑠) . 𝑠

In the following, we shall give some estimates for the extremal functions. Let 𝑁 − 𝑝 𝑝−1 ) ] 𝑝−1

𝑈𝜀 (𝑥) :=

(𝑁−𝑝)/𝑝(𝑝−𝑠)

, (34)

𝑙𝜀 (𝑥) . 𝐶𝜀

Define a function 𝜑 ∈ 𝐶0∞ (Ω) such that 𝜑(𝑥) = 1 for |𝑥| ≤ 𝑅, 𝜑(𝑥) = 0 for |𝑥| ≥ 2𝑅, 0 ≤ 𝜑(𝑥) ≤ 1, where 𝐵2𝑅 (0) ⊂ Ω. Set 𝑢𝜀 (𝑥) = 𝜑 (𝑥) 𝑈𝜀 (𝑥) , V𝜀 (𝑥) =

𝑢𝜀 (𝑥)

󵄨 󵄨𝑝∗ (𝑠) (∫Ω (󵄨󵄨󵄨𝑢𝜀 󵄨󵄨󵄨 /|𝑥|𝑠 ))

for 𝜀 󳨀→ 0+ .

(38)

Lemma 10. Suppose that 0 ≤ 𝑠 < 𝑝. If (𝐴 1 ), (𝐴 2 ) and (7) 1,𝑝 hold, then there exists 𝑢0 ∈ 𝑊0 (Ω), 𝑢0 ≢ 0, such that sup 𝐼 (𝑡𝑢0 )
0. (31)

𝑑𝑥 = 𝑘,

𝑝/𝑝∗ (𝑠) 𝑝∗ (𝑠) (V𝑛+ ) 𝑑𝑥) , 𝑠

Moreover, by using the Sobolev embedding theorem and (36), one can deduce

𝑡≥0

𝑝−𝑠 𝐴(𝑁−𝑠)/(𝑝−𝑠) 𝜇(𝑝−𝑁)/(𝑝−𝑠) . 𝑝 (𝑁 − 𝑠) 𝑠

(39)

Proof. We consider the functions ∗

𝑔 (𝑡) = 𝐼 (𝑡V𝜀 ) =

𝑡𝑝 󵄩󵄩 󵄩󵄩𝑝 𝑡𝑝 (𝑠) − ∫ 𝐹 (𝑥, 𝑡V𝜀 ) 𝑑𝑥, 󵄩󵄩V𝜀 󵄩󵄩 − 𝜇 ∗ 𝑝 𝑝 (𝑠) Ω ∗

𝑡𝑝 󵄩 󵄩𝑝 𝑡𝑝 (𝑠) 𝑔 (𝑡) = 󵄩󵄩󵄩V𝜀 󵄩󵄩󵄩 − 𝜇 ∗ . 𝑝 𝑝 (𝑠)

(40)

Since lim𝑡 → ∞ 𝑔(𝑡) = −∞, 𝑔(0) = 0, and 𝑔(𝑡) > 0 for 𝑡 small enough, sup𝑡≥0 𝑔(𝑡) is attained for some 𝑡𝜀 > 0. Therefore, we have 0 = 𝑔󸀠 (𝑡𝜀 ) ∗ 1 󵄩 󵄩𝑝 = 𝑡𝜀𝑝−1 (󵄩󵄩󵄩V𝜀 󵄩󵄩󵄩 − 𝜇𝑡𝜀𝑝 (𝑠)−𝑝 − 𝑝−1 ∫ 𝑓 (𝑥, 𝑡𝜀 V𝜀 ) V𝜀 𝑑𝑥) , Ω 𝑡𝜀

(41) and hence

1/𝑝∗ (𝑠)

,

(35)

∗ 1 󵄩󵄩 󵄩󵄩𝑝 𝑝∗ (𝑠)−𝑝 + 𝑝−1 ∫ 𝑓 (𝑥, 𝑡𝜀 V𝜀 ) V𝜀 𝑑𝑥 ≥ 𝜇𝑡𝜀𝑝 (𝑠)−𝑝 . 󵄩󵄩V𝜀 󵄩󵄩 = 𝜇𝑡𝜀 Ω 𝑡𝜀 (42)

Journal of Applied Mathematics

5

3. Proofs of Main Results

Therefore, we obtain 󵄩󵄩 󵄩󵄩𝑝 1/(𝑝 󵄩V 󵄩 𝑡𝜀 ≤ ( 󵄩 𝜀 󵄩 ) 𝜇

∗

1,𝑝

(𝑠)−𝑝)

:=

𝑡𝜀0 .

(43)

Proof of Theorem 1. Let 𝑋 := 𝑊0 (Ω). From the Sobolev and Hardy-Sobolev inequalities, we can easily get ∗

‖𝑢‖𝑝𝑝 ≤ 𝐶‖𝑢‖𝑝 ,

By (𝐴 1 ), we can easily get 󵄨 󵄨󵄨 𝑝∗ −1 + 𝑑 (𝜀) 𝑡𝑝−1 , 󵄨󵄨𝑓 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ 𝜀𝑡

𝑑 (𝜀) > 0.

∫

Ω

|𝑢|𝑝 (𝑠) 𝑝∗ (𝑠) , 𝑠 𝑑𝑥 ≤ 𝐶‖𝑢‖ |𝑥|

(44)

𝑝∗

Hence, we can get ∗ ∗ 󵄩󵄩 󵄩󵄩𝑝 𝑝∗ (𝑠)−𝑝 + 𝜀 ∫ |𝑡𝜀 |𝑝 −𝑝 |V𝜀 |𝑝 𝑑𝑥 + 𝑑 (𝜀) ∫ |V𝜀 |𝑝 𝑑𝑥. 󵄩󵄩V𝜀 󵄩󵄩 ≤ 𝜇𝑡𝜀 Ω Ω (45)

By (36)–(38), when 𝜀 is small enough, we conclude that ∗ 𝑡𝜀𝑝 (𝑠)−𝑝

𝐴 ≥ 𝑠. 2

󵄩󵄩 󵄩󵄩𝑝(𝑁−𝑠)/(𝑝−𝑠) ≤ 𝐴(𝑁−𝑠)/(𝑝−𝑠) + 𝐶7 𝜀(𝑁−𝑠)/(𝑝−𝑠) . 󵄩󵄩V𝜀 󵄩󵄩 𝑠

It follows from (𝐴 1 ) that

On the other hand, the function 𝑔(𝑡) attains its maximum at 𝑡𝜀0 and is increasing in the interval [0, 𝑡𝜀0 ]. Note that (𝐴 2 ) implies 𝐹(𝑥, 𝑡) ≥ 𝐶8 |𝑡|𝜌 , which together with (36), (46), and (47) implies that

∀𝑡 ∈ (0, 𝛿2 )

for all 𝑡 ∈ R+ and for 𝑥 ∈ Ω. Then one gets

𝑝−𝑠 𝐴(𝑁−𝑠)/(𝑝−𝑠) 𝜇(𝑝−𝑁)/(𝑝−𝑠) 𝑝 (𝑁 − 𝑠) 𝑠

− 𝐶8 (

𝐴𝑠 ) 2

(48)

(𝑢+ ) 𝜇 1 𝐼 (𝑢) = ‖𝑢‖𝑝 − ∗ ∫ 𝑝 𝑝 (𝑠) Ω |𝑥|𝑠 ≥

∗ 𝐶𝜇 󵄩 󵄩𝑝∗ (𝑠) 𝐶 1 − 𝜀‖𝑢‖𝑝 − 𝐶𝐶10 ‖𝑢‖𝑝 ‖𝑢‖𝑝 − ∗ 󵄩󵄩󵄩𝑢+ 󵄩󵄩󵄩 𝑝 𝑝 (𝑠) 𝑝 (56)

1,𝑝

𝑟 > 0 small enough.

Furthermore, from (7) and (37), we get

(50)

Therefore, by choosing 𝜀 small enough, we have sup 𝐼 (𝑡V𝜀 ) = 𝑔 (𝑡𝜀 ) < 𝑡≥0

(57)

1,𝑝

(49)

By (7), we have 𝜌 > 𝑝∗ − 𝑝/(𝑝 − 1), which implies 𝜌 (𝑁 − 𝑝) 𝑁−𝑝 𝑝−1 > (𝑁 − ). 𝑝−𝑠 𝑝−𝑠 𝑝

Ω

𝐼 (𝑢) ≥ 𝛽 ∀𝑢 ∈ 𝜕𝐵𝑟 = {𝑢 ∈ 𝑊0 (Ω) , ‖𝑢‖ = 𝑟} ,

Ω

Ω

𝑑𝑥 − ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥

for 𝜀 small enough. So there exists 𝛽 > 0 such that

∫ |V𝜀 |𝜌 𝑑𝑥.

∫ |V𝜀 |𝜌 𝑑𝑥 ≥ 𝐶4 𝜀((𝑝−1)/(𝑝−𝑠))(𝑁−𝜌(𝑁−𝑝)/𝑝) .

(55)

𝑝∗ (𝑠)

+ 𝐶9 𝜇(𝑝−𝑁)/(𝑝−𝑠) 𝜀(𝑁−𝑝)/(𝑝−𝑠) 𝜌/(𝑝∗ (𝑠)−𝑝)

∗ 1 𝑝 𝜀|𝑡| + 𝐶10 |𝑡|𝑝 𝑝

for all 𝑡 ∈ R and for 𝑥 ∈ Ω. By (52) and (55) we have

− ∫ 𝐹 (𝑥, 𝑡𝜀 V𝜀 ) 𝑑𝑥 ≤

∀𝑡 ∈ [𝛿2 , 𝛿1 ]

∗ 1−𝑝∗ 󵄨 𝑝∗ −1 󵄨󵄨 + 𝜀𝑡𝑝−1 + 𝑀 ≤ 𝜀𝑡𝑝−1 + (1 + 𝑀𝛿2 ) 𝑡𝑝 −1 󵄨󵄨𝑓 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ 𝑡 (54)

|𝐹 (𝑥, 𝑡)| ≤

𝑝 − 𝑠 󵄩󵄩 󵄩󵄩𝑝(𝑁−𝑠)/(𝑝−𝑠) (𝑝−𝑁)/(𝑝−𝑠) 𝜇 󵄩V 󵄩 𝑝 (𝑁 − 𝑠) 󵄩 𝜀 󵄩

(53)

uniformly for all 𝑥 ∈ Ω. Therefore, we deduce that

Ω

Ω

󵄨 󵄨 s.t. 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 ≤ 𝑀,

∃𝑀 > 0,

𝑔 (𝑡𝜀 ) ≤ 𝑔 (𝑡𝜀0 ) − ∫ 𝐹 (𝑥, 𝑡𝜀 V𝜀 ) 𝑑𝑥 =

∗ 󵄨 󵄨 s.t. 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 < 𝑡𝑝 −1 , ∀𝑡 > 𝛿1 ; 󵄨 󵄨 ∃0 < 𝛿2 < 𝛿1 , s.t. 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 < 𝜀𝑡𝑝−1 ,

∃𝛿1 > 0,

(46)

(47)

(52)

∀𝑢 ∈ 𝑋.

∀𝜀 > 0,

On the one hand, from Lemma 7 and (36), it follows that

∗

‖𝑢‖𝑝∗ ≤ 𝐶‖𝑢‖𝑝 ,

𝑝−𝑠 𝐴(𝑁−𝑠)/(𝑝−𝑠) 𝜇(𝑝−𝑁)/(𝑝−𝑠) . 𝑝 (𝑁 − 𝑠) 𝑠 (51)

Hence, the proof of the lemma is completed by taking 𝑢0 = V𝜀 .

By Lemma 10, there exists 𝑢0 ∈ 𝑊0 (Ω) with 𝑢0 ≢ 0 such that 𝑝−𝑠 𝐴(𝑁−𝑠)/(𝑝−𝑠) 𝜇(𝑝−𝑁)/(𝑝−𝑠) . sup 𝐼 (𝑡𝑢0 ) < (58) 𝑝 (𝑁 − 𝑠) 𝑠 𝑡≥0 It follows from the nonnegativity of 𝐹(𝑥, 𝑡) that 𝑝∗ (𝑠)

∗ (𝑢0+ ) 𝜇 1 󵄩 󵄩𝑝 𝐼 (𝑡𝑢0 ) = 𝑡𝑝 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩 − ∗ 𝑡𝑝 (𝑠) ∫ 𝑝 𝑝 (𝑠) |𝑥|𝑠 Ω

𝑑𝑥

− ∫ 𝐹 (𝑥, 𝑡𝑢0 ) 𝑑𝑥

(59)

Ω

𝑝∗ (𝑠)

∗ (𝑢0+ ) 𝜇 1 󵄩 󵄩𝑝 ≤ 𝑡𝑝 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩 − ∗ 𝑡𝑝 (𝑠) ∫ 𝑝 𝑝 (𝑠) |𝑥|𝑠 Ω

𝑑𝑥.

6

Journal of Applied Mathematics

Therefore, lim𝑡 → +∞ 𝐼(𝑡𝑢0 ) → −∞, so we can choose 𝑡0 > 0 such that 󵄩 󵄩󵄩 𝐼 (𝑡0 𝑢0 ) ≤ 0. (60) 󵄩󵄩𝑡0 𝑢0 󵄩󵄩󵄩 > 𝑟, By virtue of the Mountain Pass Lemma in [16], there is a sequence {𝑢𝑛 } ⊂ 𝑋 satisfying 𝐼󸀠 (𝑢𝑛 ) 󳨀→ 0,

𝐼 (𝑢𝑛 ) 󳨀→ 𝑐 ≥ 𝛽,

(61)

where 𝑐 = inf max 𝐼 (ℎ (𝑡)) , ℎ∈Γ 𝑡∈[0,1]

(62)

Γ = {ℎ ∈ 𝐶 ([0, 1] , 𝑋) | ℎ (0) = 0, ℎ (1) = 𝑡0 𝑢0 } . Note that 0 < 𝛽 ≤ 𝑐 = inf max 𝐼 (ℎ (𝑡)) ≤ max 𝐼 (𝑡𝑡0 𝑢0 ) ≤ sup 𝐼 (𝑡𝑢0 ) ℎ∈Γ 𝑡∈[0,1]