Quasilinear Elliptic Equations with Hardy-Sobolev Critical Exponents

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Jan 12, 2014 - Quasilinear Elliptic Equations with Hardy-Sobolev Critical. Exponents: Existence and Multiplicity of Nontrivial Solutions. Guanwei Chen.
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 Β© 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 πœ‡ = 1 hold, then (1) reduces to the quasilinear elliptic problem:

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

βˆ’Ξ”π‘ 𝑒 βˆ’ |𝑒|𝑝

βˆ—

|𝑒|𝑝 (𝑠)βˆ’2 βˆ’Ξ” 𝑝 𝑒 βˆ’ πœ‡ 𝑒 = 𝑓 (π‘₯, 𝑒) , |π‘₯|𝑠 𝑒 = 0,

π‘₯ ∈ Ξ© \ {0} ,

(1)

π‘₯ ∈ πœ•Ξ©,

where Ξ” 𝑝 𝑒 = div(|βˆ‡π‘’|π‘βˆ’2 βˆ‡π‘’) denotes the 𝑝-Laplacian differential operator, Ξ© is an open bounded domain in R𝑁 (𝑁 β‰₯ 3) with smooth boundary πœ•Ξ© and 0 ∈ Ξ©, 0 ≀ 𝑠 < 𝑝, π‘βˆ— (𝑠) = 𝑝(𝑁 βˆ’ 𝑠)/(𝑁 βˆ’ 𝑝) is the Hardy-Sobolev critical exponent, and π‘βˆ— = π‘βˆ— (0) = 𝑁𝑝/(𝑁 βˆ’ 𝑝) is the Sobolev critical exponent. Here, we let ‖𝑒‖ := (∫ |βˆ‡π‘’|𝑝 𝑑π‘₯)

1/𝑝

Ξ©

,

(2)

which is equivalent to the usual norm of Sobolev space 1,𝑝 π‘Š0 (Ξ©) due to the PoincarΒ΄e inequality. Let 𝐴 𝑠 (Ξ©) :=

inf 1,𝑝

‖𝑒‖𝑝 π‘βˆ— (𝑠)

π‘’βˆˆπ‘Š0 (Ξ©)\{0} (∫ (|𝑒| Ξ©

/|π‘₯|𝑠 )𝑑π‘₯)

which is the best Hardy-Sobolev constant.

𝑝/π‘βˆ— (𝑠)

,

(3)

βˆ—

βˆ’2

𝑒 = 𝑓 (π‘₯, 𝑒) ,

𝑒 = 0,

π‘₯ ∈ πœ•Ξ©.

π‘₯ ∈ Ξ©,

(4)

GoncΒΈalves and Alves [1] have studied (4) in R𝑁 involving 𝑓(π‘₯, 𝑒) = β„Ž(π‘₯)π‘’π‘ž , 𝑒 β‰₯ 0 and 𝑒 ≑̸ 0 to obtain existence of positive solutions where 2 ≀ 𝑝 < 𝑁, 0 < π‘ž < 𝑝 βˆ’ 1, or 𝑝 βˆ’ 1 < π‘ž < π‘βˆ— βˆ’ 1 and a suitable β„Ž. We should mention that problem (4) with 𝑝 = 2 has been widely studied since BrΒ΄ezis and Nirenberg; see [2–4] and the references therein. Ghoussoub and Yuan [5] have studied (1) with 𝑓(π‘₯, 𝑒) = πœ†|𝑒|π‘Ÿβˆ’2 𝑒, where 𝑝 < π‘Ÿ < π‘βˆ— . They obtained a positive solution in the case where πœ† > 0, πœ‡ > 0, and 𝑁 > [𝑝(𝑝 βˆ’ 1)π‘Ÿ + 𝑝2 ]/[𝑝 + (𝑝 βˆ’ 1)(π‘Ÿ βˆ’ 𝑝)] (in particular if 𝑁 β‰₯ 𝑝2 ) hold. They also obtained a sign-changing solution in the case where πœ† > 0, πœ‡ > 0, and 𝑁 > [𝑝(𝑝 βˆ’ 1)π‘Ÿ + 𝑝]/[1 + (𝑝 βˆ’ 1)(π‘Ÿ βˆ’ 𝑝)] (in particular if 𝑁 > 𝑝3 βˆ’ 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 π‘Š0 (Ξ©), which is not a Hilbert space for 𝑝 =ΜΈ 2. Due to the lack of compactness of the embedding βˆ— βˆ— 1,𝑝 1,𝑝 in π‘Š0 (Ξ©) 󳨅→ 𝐿𝑝 (Ξ©) and π‘Š0 (Ξ©) 󳨅→ 𝐿𝑝 (𝑠) (Ξ©, |π‘₯|βˆ’π‘  𝑑π‘₯), 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 π‘Š0 (Ξ©). However,

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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(π‘₯, 0) ≑ 0. 𝑑 Let 𝐹(π‘₯, 𝑑) := ∫0 𝑓(π‘₯, 𝑠)𝑑𝑠, π‘₯ ∈ Ξ©. To state our main results, we still need the following assumptions. (𝐴 1 ) 𝑓 ∈ 𝐢(Ξ© Γ— R+ , R), lim𝑑 β†’ 0+ (𝑓(π‘₯, 𝑑)/π‘‘π‘βˆ’1 ) = 0, and βˆ— lim𝑑 β†’ ∞ (𝑓(π‘₯, 𝑑)/𝑑𝑝 βˆ’1 ) = 0 uniformly for π‘₯ ∈ Ξ©. (𝐴 2 ) There exists a constant 𝜌 with 𝜌 > 𝑝 such that 0 < 𝜌𝐹 (π‘₯, 𝑑) ≀ 𝑓 (π‘₯, 𝑑) 𝑑,

βˆ€π‘₯ ∈ Ξ©, βˆ€π‘‘ ∈ 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 β€– β‹… ‖𝑝 denote the norm in 𝐿𝑝 (Ξ©). It is obvious that the values of 𝑓(π‘₯, 𝑑) for 𝑑 < 0 are irrelevant in Theorem 1, so we may define 𝑓 (π‘₯, 𝑑) ≑ 0

βˆ€π‘₯ ∈ Ξ©, βˆ€π‘‘ ∈ R \ {0} .

(5)

(6)

π‘βˆ— (𝑠)βˆ’1

(𝑒+ ) βˆ’Ξ” 𝑝 𝑒 = πœ‡ |π‘₯|𝑠

(10)

π‘₯ ∈ πœ•Ξ©.

π‘βˆ— (𝑠)

𝑝 1 ) , π‘βˆ— βˆ’ } 𝑝 π‘βˆ’1

(𝑒+ ) πœ‡ 1 𝐼 (𝑒) = ∫ |βˆ‡π‘’|𝑝 𝑑π‘₯ βˆ’ βˆ— ∫ 𝑝 Ξ© 𝑝 (𝑠) Ξ© |π‘₯|𝑠 βˆ’ ∫ 𝐹 (π‘₯, 𝑒) 𝑑π‘₯, Ξ©

(7)

hold, then (1) has at least one positive solution. Theorem 2. Suppose that 𝑁 β‰₯ 3, 0 < πœ‡ < ∞, 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 < πœ‡ < ∞, 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 < πœ‡ < ∞, 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(π‘₯, 𝑒) = πœ†|𝑒|π‘Ÿβˆ’2 𝑒, 𝑝 < π‘Ÿ < π‘βˆ— . There are functions 𝑓 satisfying the assumptions of our Theorem 1 and not satisfying those in [5]. Let (π‘₯, 𝑑) ∈ Ξ© Γ— R,

(8)

𝑑π‘₯ (11)

1,𝑝

𝑒 ∈ π‘Š0 (Ξ©) .

By Hardy-Sobolev inequalities (see [5, 14]) and (𝐴 1 ), we know 1,𝑝 𝐼 ∈ 𝐢1 (π‘Š0 (Ξ©), 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 π‘Š0 (Ξ©). More precisely 1,𝑝 we say that 𝑒 ∈ π‘Š0 (Ξ©) is a weak solution of (10), if, for any 1,𝑝 V ∈ π‘Š0 (Ξ©), there holds σΈ€ 

π‘βˆ’2

⟨𝐼 (𝑒) , V⟩ = ∫ |βˆ‡π‘’| Ξ©

π‘βˆ— (𝑠)βˆ’1

(𝑒+ ) (βˆ‡π‘’, βˆ‡V) 𝑑π‘₯ βˆ’ πœ‡ ∫ |π‘₯|𝑠 Ξ©

V𝑑π‘₯

βˆ’ ∫ 𝑓 (π‘₯, 𝑒) V𝑑π‘₯ = 0. Ξ©

2

∞

π‘₯ ∈ Ξ© \ {0} ,

The energy functional corresponding to (10) is given by

Theorem 1. Suppose that 𝑁 β‰₯ 3, 0 < πœ‡ < ∞, 0 ≀ 𝑠 < 𝑝, and 1 < 𝑝 < 𝑁 hold. If (𝐴 1 ), (𝐴 2 ), and

𝑓 (π‘₯, 𝑑) := 𝑔 (π‘₯) |𝑑|π‘˜βˆ’2 𝑑 + π‘Ž|𝑑|π‘™βˆ’2 𝑑,

+ 𝑓 (π‘₯, 𝑒) ,

𝑒 = 0,

Now, our main results read as follows.

𝜌 > max {𝑝, π‘βˆ— (1 βˆ’

(9)

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

(𝐴 3 ) 𝑓 ∈ 𝐢(Ξ© Γ— R, R), lim𝑑 β†’ 0 (𝑓(π‘₯, 𝑑)/|𝑑|π‘βˆ’1 ) = 0, and βˆ— lim|𝑑| β†’ ∞ (𝑓(π‘₯, 𝑑)/|𝑑|𝑝 βˆ’1 ) = 0 uniformly for π‘₯ ∈ Ξ©. (𝐴 4 ) There exists a constant 𝜌 with 𝜌 > 𝑝 such that 0 < 𝜌𝐹 (π‘₯, 𝑑) ≀ 𝑓 (π‘₯, 𝑑) 𝑑,

for π‘₯ ∈ Ξ©, 𝑑 ≀ 0.

(12)

Lemma 6 (see [15]). If 𝑓𝑛 β†’ 𝑓 a.e. in Ξ© and ‖𝑓𝑛 ‖𝑝 ≀ 𝐢 < ∞ for all n and some 0 < 𝑝 < ∞, then σ΅„© 󡄩𝑝 σ΅„© 󡄩𝑝 σ΅„© 󡄩𝑝 lim (󡄩󡄩󡄩𝑓𝑛 󡄩󡄩󡄩𝑝 βˆ’ 󡄩󡄩󡄩𝑓𝑛 βˆ’ 𝑓󡄩󡄩󡄩𝑝 ) = 󡄩󡄩󡄩𝑓󡄩󡄩󡄩𝑝 .

π‘›β†’βˆž

(13)

Lemma 7. For any π‘Ž > 0, 0 ≀ 𝑏 ≀ 1, and πœ† β‰₯ 1, we have (π‘Ž + 𝑏)πœ† ≀ π‘Žπœ† + πœ†(π‘Ž + 1)πœ†βˆ’1 𝑏. Proof. Let

βˆ—

where 𝑔(π‘₯) > 0, 𝑔 ∈ 𝐿 (Ξ©), π‘Ž > 0, and 𝑝 < π‘˜ < 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].

β„Ž (π‘₯) := (π‘Ž + π‘₯)πœ† βˆ’ π‘Žπœ† βˆ’ πœ†(π‘Ž + 1)πœ†βˆ’1 π‘₯. Clearly, β„ŽσΈ€  (π‘₯) < 0, π‘₯ ∈ (0, 1), so β„Ž(𝑏) ≀ β„Ž(0) = 0.

(14)

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3 π‘βˆ—

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

By the continuity of embedding, we have ‖𝑒𝑛 β€–π‘βˆ— ≀ 𝐢1 < ∞. From [5], going if necessary to a subsequence, one can get that βˆ‡π‘’π‘› 󳨀→ βˆ‡π‘’ a.e. in Ξ©, 𝑒𝑛 𝑒 ⇀ π‘₯ π‘₯

(ii) 𝐴 𝑠 (Ξ©) is attained when Ξ© = R𝑁 by the functions (π‘βˆ’π‘)/𝑝(π‘βˆ’π‘ )

𝑁 βˆ’ 𝑝 π‘βˆ’1 π‘™πœ€ (π‘₯) = (πœ€ (𝑁 βˆ’ 𝑠) ( ) ) π‘βˆ’1

(15)

weakly in 𝐿𝑝 (Ξ©) ,

βˆ— 󡄨󡄨 σ΅„¨σ΅„¨π‘βˆ— (𝑠)βˆ’2 𝑒 󡄨𝑒 󡄨 |𝑒|𝑝 (𝑠)βˆ’2 𝑒 V𝑑π‘₯, ∫ 󡄨 𝑛 󡄨 𝑠 𝑛 V𝑑π‘₯ 󳨀→ ∫ |π‘₯| |π‘₯|𝑠 Ξ© Ξ©

1,𝑝

βˆ€V ∈ π‘Š0 (Ξ©) (20)

(π‘βˆ’π‘ )/(π‘βˆ’1) (π‘βˆ’π‘)/(π‘βˆ’π‘ )

Γ— (πœ€ + |π‘₯|

)

for some πœ€ > 0. Moreover, the functions π‘™πœ€ (π‘₯) are the only positive radial solutions of βˆ—

𝑒𝑝 (𝑠)βˆ’1 βˆ’Ξ” 𝑝 𝑒 = |π‘₯|𝑠

(16)

∫

R𝑁

󡄨󡄨 󡄨󡄨𝑝 σ΅„¨σ΅„¨βˆ‡π‘™πœ€ 󡄨󡄨 𝑑π‘₯ = ∫ R𝑁

βˆ— 1 󡄨 󡄨󡄨 πœ€|𝑑|𝑝 󡄨󡄨𝑓 (π‘₯, 𝑑) 𝑑󡄨󡄨󡄨 ≀ π‘Ž (πœ€) + 2𝐢1

for (π‘₯, 𝑑) ∈ Ξ© Γ— (0, +∞) . (21)

Set 𝛿 := πœ€/2π‘Ž(πœ€) > 0. When 𝐸 βŠ‚ Ξ©, meas(𝐸) < 𝛿, we get 󡄨 󡄨󡄨 βˆ— σ΅„¨σ΅„¨βˆ« 𝑓 (π‘₯, 𝑒 ) 𝑒 𝑑π‘₯󡄨󡄨󡄨 ≀ ∫ π‘Ž (πœ€) 𝑑π‘₯ + πœ€ ∫ 󡄨󡄨󡄨𝑒 󡄨󡄨󡄨𝑝 𝑑π‘₯ 󡄨󡄨 󡄨󡄨 𝑛 𝑛 𝑛󡄨 󡄨 2𝐢1 𝐸 󡄨 󡄨 𝐸 𝐸

in R𝑁, and satisfy 󡄨󡄨 σ΅„¨σ΅„¨π‘βˆ— (𝑠) σ΅„¨σ΅„¨π‘™πœ€ 󡄨󡄨 (π‘βˆ’π‘ )/(π‘βˆ’π‘ ) 𝑑π‘₯ = (𝐴 𝑠 (Ξ©)) . |π‘₯|𝑠

as 𝑛 β†’ ∞. By (𝐴 1 ), we know that for any πœ€ > 0 there exists π‘Ž(πœ€) > 0 such that

(22)

< πœ€ 󳨀→ 0 as meas (𝐸) 󳨀→ 0.

(17)

It follows from Vitali’s theorem that Lemma 9. If (𝐴 1 ), (𝐴 2 ), and 0 < 𝑐 < ((𝑝 βˆ’ 𝑠)/𝑝(𝑁 βˆ’ 𝑠))𝐴(π‘βˆ’π‘ )/(π‘βˆ’π‘ ) πœ‡(π‘βˆ’π‘)/(π‘βˆ’π‘ ) hold, then 𝐼 satisfies (PS)𝑐 condition. 𝑠 1,𝑝

Proof. Suppose that {𝑒𝑛 } is a (PS)𝑐 sequence in π‘Š0 (Ξ©). By (𝐴 2 ), we have

Ξ©

1 σΈ€  ⟨𝐼 (𝑒𝑛 ) , 𝑒𝑛 ⟩ πœƒ

as 𝑛 󳨀→ ∞. (23)

Similarly, we can also get

Ξ©

Ξ©

as 𝑛 󳨀→ ∞.

(24)

Since 𝐼󸀠 (𝑒𝑛 ) β†’ 0, we have π‘βˆ— (𝑠)

(𝑒𝑛+ ) 1 1 1 σ΅„© 󡄩𝑝 1 = ( βˆ’ ) 󡄩󡄩󡄩𝑒𝑛 σ΅„©σ΅„©σ΅„© + ( βˆ’ βˆ— ) πœ‡ ∫ 𝑝 πœƒ πœƒ 𝑝 (𝑠) |π‘₯|𝑠 Ξ©

𝑑π‘₯

1 βˆ’ ∫ (𝐹 (π‘₯, 𝑒𝑛 ) βˆ’ 𝑓 (π‘₯, 𝑒𝑛 ) 𝑒𝑛 ) 𝑑π‘₯ πœƒ Ξ© β‰₯(

Ξ©

∫ 𝐹 (π‘₯, 𝑒𝑛 ) 𝑑π‘₯ 󳨀→ ∫ 𝐹 (π‘₯, 𝑒) 𝑑π‘₯

σ΅„© σ΅„© 𝑐 + 1 + π‘œ (1) 󡄩󡄩󡄩𝑒𝑛 σ΅„©σ΅„©σ΅„© β‰₯ 𝐼 (𝑒𝑛 ) βˆ’

∫ 𝑓 (π‘₯, 𝑒𝑛 ) 𝑒𝑛 𝑑π‘₯ 󳨀→ ∫ 𝑓 (π‘₯, 𝑒) 𝑒 𝑑π‘₯

1 1 σ΅„©σ΅„© 󡄩󡄩𝑝 βˆ’ ) 󡄩𝑒 σ΅„© , 𝑝 πœƒ σ΅„© 𝑛󡄩

βŸ¨πΌσΈ€  (𝑒𝑛 ) , 𝑒𝑛 ⟩ π‘βˆ— (𝑠)

(𝑒𝑛+ ) σ΅„© 󡄩𝑝 = 󡄩󡄩󡄩𝑒𝑛 σ΅„©σ΅„©σ΅„© βˆ’ πœ‡ ∫ |π‘₯|𝑠 Ξ©

𝑑π‘₯ βˆ’ ∫ 𝑓 (π‘₯, 𝑒𝑛 ) 𝑒𝑛 𝑑π‘₯ = π‘œ (1) . Ξ©

(25)

Let V𝑛 := 𝑒𝑛 βˆ’ 𝑒, which together with Lemma 6 implies (18)

βˆ—

where πœƒ = min{𝜌, 𝑝 (𝑠)}. Hence we conclude that {𝑒𝑛 } is a 1,𝑝 1,𝑝 bounded sequence in π‘Š0 (Ξ©). So there exists 𝑒 ∈ π‘Š0 (Ξ©); going if necessary to a subsequence, we have

(1 < 𝛾 < π‘βˆ— ) ,

𝑒𝑛 󳨀→ 𝑒 a.e. in Ξ©, 𝑛 󳨀→ ∞.

π‘βˆ— (𝑠)

(𝑒+ ) 𝑑π‘₯ βˆ’ πœ‡ ∫ |π‘₯|𝑠 Ξ©

𝑑π‘₯ (26)

βˆ’ ∫ 𝑓 (π‘₯, 𝑒) 𝑒 𝑑π‘₯ = π‘œ (1) . Ξ©

From (20), we can obtain

1,𝑝

𝑒𝑛 ⇀ 𝑒 in π‘Š0 (Ξ©) , 𝑒𝑛 󳨀→ 𝑒 in 𝐿𝛾 (Ξ©)

π‘βˆ— (𝑠)

(V𝑛+ ) σ΅„©σ΅„© 󡄩󡄩𝑝 𝑝 σ΅„©σ΅„©V𝑛 σ΅„©σ΅„© + ‖𝑒‖ βˆ’ πœ‡ ∫ |π‘₯|𝑠 Ξ©

(19)

π‘βˆ— (𝑠)

(𝑒+ ) lim ⟨𝐼 (𝑒𝑛 ) , π‘’βŸ© = ‖𝑒‖ βˆ’ πœ‡ ∫ π‘›β†’βˆž |π‘₯|𝑠 Ξ© σΈ€ 

𝑝

𝑑π‘₯

βˆ’ ∫ 𝑓 (π‘₯, 𝑒) 𝑒 𝑑π‘₯ = 0. Ξ©

(27)

4

Journal of Applied Mathematics

Note that 𝐼(𝑒𝑛 ) β†’ 𝑐 as 𝑛 β†’ ∞, which together with Lemma 6 implies π‘βˆ— (𝑠)

(V𝑛+ ) πœ‡ 1 σ΅„© 󡄩𝑝 1 𝐼 (𝑒𝑛 ) = σ΅„©σ΅„©σ΅„©V𝑛 σ΅„©σ΅„©σ΅„© + ‖𝑒‖𝑝 βˆ’ βˆ— ∫ 𝑝 𝑝 𝑝 (𝑠) Ξ© |π‘₯|𝑠 βˆ’

βˆ— + 𝑝 (𝑠)

(𝑒 ) πœ‡ ∫ (𝑠) Ξ© |π‘₯|𝑠

π‘βˆ—

βˆ—

so that ∫Ω (|Vπœ€ |𝑝 (𝑠) /|π‘₯|𝑠 ) = 1. Then, by using the argument as used in [5], we can get the following results: σ΅„© 󡄩𝑝 𝐴 𝑠 + 𝐢2 πœ€(π‘βˆ’π‘)/(π‘βˆ’π‘ ) ≀ σ΅„©σ΅„©σ΅„©Vπœ€ σ΅„©σ΅„©σ΅„© ≀ 𝐴 𝑠 + 𝐢3 πœ€(π‘βˆ’π‘)/(π‘βˆ’π‘ ) ,

𝑑π‘₯

𝑑π‘₯ βˆ’ ∫ 𝐹 (π‘₯, 𝑒) 𝑑π‘₯ + π‘œ (1)

󡄨 σ΅„¨π‘Ÿ 𝐢4 πœ€(π‘βˆ’π‘)π‘Ÿ/𝑝(π‘βˆ’π‘ ) ≀ ∫ 󡄨󡄨󡄨Vπœ€ 󡄨󡄨󡄨 𝑑π‘₯ ≀ 𝐢5 πœ€(π‘βˆ’π‘)π‘Ÿ/𝑝(π‘βˆ’π‘ ) , Ξ©

Ξ©

1 ≀ π‘Ÿ < π‘βˆ— (1 βˆ’

βˆ—

𝑝 (𝑠)

= 𝐼 (𝑒) +

(V𝑛+ ) πœ‡ 1 σ΅„©σ΅„© 󡄩󡄩𝑝 ∫ σ΅„©σ΅„©V𝑛 σ΅„©σ΅„© βˆ’ βˆ— 𝑝 𝑝 (𝑠) Ξ© |π‘₯|𝑠

𝑑π‘₯ + π‘œ (1)

= 𝑐 + π‘œ (1) .

π‘Ÿ = π‘βˆ— (1 βˆ’

Therefore, one gets that

1 ), 𝑝

𝐢4 πœ€((π‘βˆ’1)/(π‘βˆ’π‘ ))(π‘βˆ’π‘Ÿ(π‘βˆ’π‘)/𝑝)

π‘βˆ— (𝑠)

𝑑π‘₯ = 𝑐 + π‘œ (1) .

(29)

󡄨 σ΅„¨π‘Ÿ ≀ ∫ 󡄨󡄨󡄨Vπœ€ 󡄨󡄨󡄨 𝑑π‘₯ ≀ 𝐢5 πœ€((π‘βˆ’1)/(π‘βˆ’π‘ ))(π‘βˆ’π‘Ÿ(π‘βˆ’π‘)/𝑝) , Ξ©

From (26) and (27), we have

π‘βˆ— (1 βˆ’

π‘βˆ— (𝑠)

(V𝑛+ ) σ΅„©σ΅„© 󡄩󡄩𝑝 σ΅„©σ΅„©V𝑛 σ΅„©σ΅„© βˆ’ πœ‡ ∫ |π‘₯|𝑠 Ξ©

1 ), 𝑝

󡄨 σ΅„¨π‘Ÿ 𝐢4 πœ€(π‘βˆ’π‘)π‘Ÿ/𝑝(π‘βˆ’π‘ ) |ln πœ€| ≀ ∫ 󡄨󡄨󡄨Vπœ€ 󡄨󡄨󡄨 𝑑π‘₯ ≀ 𝐢5 πœ€(π‘βˆ’π‘)π‘Ÿ/𝑝(π‘βˆ’π‘ ) |ln πœ€| , Ξ©

(28) (V𝑛+ ) πœ‡ 1 σ΅„© 󡄩𝑝 𝐼 (𝑒) + σ΅„©σ΅„©σ΅„©V𝑛 σ΅„©σ΅„©σ΅„© βˆ’ βˆ— ∫ 𝑝 𝑝 (𝑠) Ξ© |π‘₯|𝑠

(36)

(30)

𝑑π‘₯ = π‘œ (1) ;

1 ) < π‘Ÿ < π‘βˆ— . 𝑝 (37)

𝑝

then β€–V𝑛 β€– β†’ 0 as 𝑛 β†’ ∞. Otherwise, there exists a subsequence (still denoted by V𝑛 ) such that π‘βˆ— (𝑠)

(V𝑛+ ) lim πœ‡ ∫ π‘›β†’βˆž |π‘₯|𝑠 Ξ©

σ΅„© 󡄩𝑝 lim σ΅„©σ΅„©V σ΅„©σ΅„© = π‘˜, 𝑛 β†’ βˆžσ΅„© 𝑛 σ΅„© By (3), we have σ΅„©σ΅„© 󡄩󡄩𝑝 σ΅„©σ΅„©V𝑛 σ΅„©σ΅„© β‰₯ 𝐴 𝑠 (∫

Ξ©

|π‘₯|

𝑝/π‘βˆ— (𝑠)

βˆ€π‘› ∈ N;

(32)

𝐴(π‘βˆ’π‘ )/(π‘βˆ’π‘ ) πœ‡(π‘βˆ’π‘)/(π‘βˆ’π‘ ) . 𝑠

In the following, we shall give some estimates for the extremal functions. Let 𝑁 βˆ’ 𝑝 π‘βˆ’1 ) ] π‘βˆ’1

π‘ˆπœ€ (π‘₯) :=

(π‘βˆ’π‘)/𝑝(π‘βˆ’π‘ )

, (34)

π‘™πœ€ (π‘₯) . πΆπœ€

Define a function πœ‘ ∈ 𝐢0∞ (Ξ©) such that πœ‘(π‘₯) = 1 for |π‘₯| ≀ 𝑅, πœ‘(π‘₯) = 0 for |π‘₯| β‰₯ 2𝑅, 0 ≀ πœ‘(π‘₯) ≀ 1, where 𝐡2𝑅 (0) βŠ‚ Ξ©. Set π‘’πœ€ (π‘₯) = πœ‘ (π‘₯) π‘ˆπœ€ (π‘₯) , Vπœ€ (π‘₯) =

π‘’πœ€ (π‘₯)

󡄨 σ΅„¨π‘βˆ— (𝑠) (∫Ω (σ΅„¨σ΅„¨σ΅„¨π‘’πœ€ 󡄨󡄨󡄨 /|π‘₯|𝑠 ))

for πœ€ 󳨀→ 0+ .

(38)

Lemma 10. Suppose that 0 ≀ 𝑠 < 𝑝. If (𝐴 1 ), (𝐴 2 ) and (7) 1,𝑝 hold, then there exists 𝑒0 ∈ π‘Š0 (Ξ©), 𝑒0 ≑̸ 0, such that sup 𝐼 (𝑑𝑒0 )
0. (31)

𝑑π‘₯ = π‘˜,

𝑝/π‘βˆ— (𝑠) π‘βˆ— (𝑠) (V𝑛+ ) 𝑑π‘₯) , 𝑠

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

𝑑β‰₯0

π‘βˆ’π‘  𝐴(π‘βˆ’π‘ )/(π‘βˆ’π‘ ) πœ‡(π‘βˆ’π‘)/(π‘βˆ’π‘ ) . 𝑝 (𝑁 βˆ’ 𝑠) 𝑠

(39)

Proof. We consider the functions βˆ—

𝑔 (𝑑) = 𝐼 (𝑑Vπœ€ ) =

𝑑𝑝 σ΅„©σ΅„© 󡄩󡄩𝑝 𝑑𝑝 (𝑠) βˆ’ ∫ 𝐹 (π‘₯, 𝑑Vπœ€ ) 𝑑π‘₯, σ΅„©σ΅„©Vπœ€ σ΅„©σ΅„© βˆ’ πœ‡ βˆ— 𝑝 𝑝 (𝑠) Ξ© βˆ—

𝑑𝑝 σ΅„© 󡄩𝑝 𝑑𝑝 (𝑠) 𝑔 (𝑑) = σ΅„©σ΅„©σ΅„©Vπœ€ σ΅„©σ΅„©σ΅„© βˆ’ πœ‡ βˆ— . 𝑝 𝑝 (𝑠)

(40)

Since lim𝑑 β†’ ∞ 𝑔(𝑑) = βˆ’βˆž, 𝑔(0) = 0, and 𝑔(𝑑) > 0 for 𝑑 small enough, sup𝑑β‰₯0 𝑔(𝑑) is attained for some π‘‘πœ€ > 0. Therefore, we have 0 = 𝑔󸀠 (π‘‘πœ€ ) βˆ— 1 σ΅„© 󡄩𝑝 = π‘‘πœ€π‘βˆ’1 (σ΅„©σ΅„©σ΅„©Vπœ€ σ΅„©σ΅„©σ΅„© βˆ’ πœ‡π‘‘πœ€π‘ (𝑠)βˆ’π‘ βˆ’ π‘βˆ’1 ∫ 𝑓 (π‘₯, π‘‘πœ€ Vπœ€ ) Vπœ€ 𝑑π‘₯) , Ξ© π‘‘πœ€

(41) and hence

1/π‘βˆ— (𝑠)

,

(35)

βˆ— 1 σ΅„©σ΅„© 󡄩󡄩𝑝 π‘βˆ— (𝑠)βˆ’π‘ + π‘βˆ’1 ∫ 𝑓 (π‘₯, π‘‘πœ€ Vπœ€ ) Vπœ€ 𝑑π‘₯ β‰₯ πœ‡π‘‘πœ€π‘ (𝑠)βˆ’π‘ . σ΅„©σ΅„©Vπœ€ σ΅„©σ΅„© = πœ‡π‘‘πœ€ Ξ© π‘‘πœ€ (42)

Journal of Applied Mathematics

5

3. Proofs of Main Results

Therefore, we obtain σ΅„©σ΅„© 󡄩󡄩𝑝 1/(𝑝 σ΅„©V σ΅„© π‘‘πœ€ ≀ ( σ΅„© πœ€ σ΅„© ) πœ‡

βˆ—

1,𝑝

(𝑠)βˆ’π‘)

:=

π‘‘πœ€0 .

(43)

Proof of Theorem 1. Let 𝑋 := π‘Š0 (Ξ©). From the Sobolev and Hardy-Sobolev inequalities, we can easily get βˆ—

‖𝑒‖𝑝𝑝 ≀ 𝐢‖𝑒‖𝑝 ,

By (𝐴 1 ), we can easily get 󡄨 󡄨󡄨 π‘βˆ— βˆ’1 + 𝑑 (πœ€) π‘‘π‘βˆ’1 , 󡄨󡄨𝑓 (π‘₯, 𝑑)󡄨󡄨󡄨 ≀ πœ€π‘‘

𝑑 (πœ€) > 0.

∫

Ξ©

|𝑒|𝑝 (𝑠) π‘βˆ— (𝑠) , 𝑠 𝑑π‘₯ ≀ 𝐢‖𝑒‖ |π‘₯|

(44)

π‘βˆ—

Hence, we can get βˆ— βˆ— σ΅„©σ΅„© 󡄩󡄩𝑝 π‘βˆ— (𝑠)βˆ’π‘ + πœ€ ∫ |π‘‘πœ€ |𝑝 βˆ’π‘ |Vπœ€ |𝑝 𝑑π‘₯ + 𝑑 (πœ€) ∫ |Vπœ€ |𝑝 𝑑π‘₯. σ΅„©σ΅„©Vπœ€ σ΅„©σ΅„© ≀ πœ‡π‘‘πœ€ Ξ© Ξ© (45)

By (36)–(38), when πœ€ is small enough, we conclude that βˆ— π‘‘πœ€π‘ (𝑠)βˆ’π‘

𝐴 β‰₯ 𝑠. 2

σ΅„©σ΅„© 󡄩󡄩𝑝(π‘βˆ’π‘ )/(π‘βˆ’π‘ ) ≀ 𝐴(π‘βˆ’π‘ )/(π‘βˆ’π‘ ) + 𝐢7 πœ€(π‘βˆ’π‘ )/(π‘βˆ’π‘ ) . σ΅„©σ΅„©Vπœ€ σ΅„©σ΅„© 𝑠

It follows from (𝐴 1 ) that

On the other hand, the function 𝑔(𝑑) attains its maximum at π‘‘πœ€0 and is increasing in the interval [0, π‘‘πœ€0 ]. Note that (𝐴 2 ) implies 𝐹(π‘₯, 𝑑) β‰₯ 𝐢8 |𝑑|𝜌 , which together with (36), (46), and (47) implies that

βˆ€π‘‘ ∈ (0, 𝛿2 )

for all 𝑑 ∈ R+ and for π‘₯ ∈ Ξ©. Then one gets

π‘βˆ’π‘  𝐴(π‘βˆ’π‘ )/(π‘βˆ’π‘ ) πœ‡(π‘βˆ’π‘)/(π‘βˆ’π‘ ) 𝑝 (𝑁 βˆ’ 𝑠) 𝑠

βˆ’ 𝐢8 (

𝐴𝑠 ) 2

(48)

(𝑒+ ) πœ‡ 1 𝐼 (𝑒) = ‖𝑒‖𝑝 βˆ’ βˆ— ∫ 𝑝 𝑝 (𝑠) Ξ© |π‘₯|𝑠 β‰₯

βˆ— πΆπœ‡ σ΅„© σ΅„©π‘βˆ— (𝑠) 𝐢 1 βˆ’ πœ€β€–π‘’β€–π‘ βˆ’ 𝐢𝐢10 ‖𝑒‖𝑝 ‖𝑒‖𝑝 βˆ’ βˆ— 󡄩󡄩󡄩𝑒+ σ΅„©σ΅„©σ΅„© 𝑝 𝑝 (𝑠) 𝑝 (56)

1,𝑝

π‘Ÿ > 0 small enough.

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

(50)

Therefore, by choosing πœ€ small enough, we have sup 𝐼 (𝑑Vπœ€ ) = 𝑔 (π‘‘πœ€ ) < 𝑑β‰₯0

(57)

1,𝑝

(49)

By (7), we have 𝜌 > π‘βˆ— βˆ’ 𝑝/(𝑝 βˆ’ 1), which implies 𝜌 (𝑁 βˆ’ 𝑝) π‘βˆ’π‘ π‘βˆ’1 > (𝑁 βˆ’ ). π‘βˆ’π‘  π‘βˆ’π‘  𝑝

Ξ©

𝐼 (𝑒) β‰₯ 𝛽 βˆ€π‘’ ∈ πœ•π΅π‘Ÿ = {𝑒 ∈ π‘Š0 (Ξ©) , ‖𝑒‖ = π‘Ÿ} ,

Ξ©

Ξ©

𝑑π‘₯ βˆ’ ∫ 𝐹 (π‘₯, 𝑒) 𝑑π‘₯

for πœ€ small enough. So there exists 𝛽 > 0 such that

∫ |Vπœ€ |𝜌 𝑑π‘₯.

∫ |Vπœ€ |𝜌 𝑑π‘₯ β‰₯ 𝐢4 πœ€((π‘βˆ’1)/(π‘βˆ’π‘ ))(π‘βˆ’πœŒ(π‘βˆ’π‘)/𝑝) .

(55)

π‘βˆ— (𝑠)

+ 𝐢9 πœ‡(π‘βˆ’π‘)/(π‘βˆ’π‘ ) πœ€(π‘βˆ’π‘)/(π‘βˆ’π‘ ) 𝜌/(π‘βˆ— (𝑠)βˆ’π‘)

βˆ— 1 𝑝 πœ€|𝑑| + 𝐢10 |𝑑|𝑝 𝑝

for all 𝑑 ∈ R and for π‘₯ ∈ Ξ©. By (52) and (55) we have

βˆ’ ∫ 𝐹 (π‘₯, π‘‘πœ€ Vπœ€ ) 𝑑π‘₯ ≀

βˆ€π‘‘ ∈ [𝛿2 , 𝛿1 ]

βˆ— 1βˆ’π‘βˆ— 󡄨 π‘βˆ— βˆ’1 󡄨󡄨 + πœ€π‘‘π‘βˆ’1 + 𝑀 ≀ πœ€π‘‘π‘βˆ’1 + (1 + 𝑀𝛿2 ) 𝑑𝑝 βˆ’1 󡄨󡄨𝑓 (π‘₯, 𝑑)󡄨󡄨󡄨 ≀ 𝑑 (54)

|𝐹 (π‘₯, 𝑑)| ≀

𝑝 βˆ’ 𝑠 σ΅„©σ΅„© 󡄩󡄩𝑝(π‘βˆ’π‘ )/(π‘βˆ’π‘ ) (π‘βˆ’π‘)/(π‘βˆ’π‘ ) πœ‡ σ΅„©V σ΅„© 𝑝 (𝑁 βˆ’ 𝑠) σ΅„© πœ€ σ΅„©

(53)

uniformly for all π‘₯ ∈ Ξ©. Therefore, we deduce that

Ξ©

Ξ©

󡄨 󡄨 s.t. 󡄨󡄨󡄨𝑓 (𝑑)󡄨󡄨󡄨 ≀ 𝑀,

βˆƒπ‘€ > 0,

𝑔 (π‘‘πœ€ ) ≀ 𝑔 (π‘‘πœ€0 ) βˆ’ ∫ 𝐹 (π‘₯, π‘‘πœ€ Vπœ€ ) 𝑑π‘₯ =

βˆ— 󡄨 󡄨 s.t. 󡄨󡄨󡄨𝑓 (𝑑)󡄨󡄨󡄨 < 𝑑𝑝 βˆ’1 , βˆ€π‘‘ > 𝛿1 ; 󡄨 󡄨 βˆƒ0 < 𝛿2 < 𝛿1 , s.t. 󡄨󡄨󡄨𝑓 (𝑑)󡄨󡄨󡄨 < πœ€π‘‘π‘βˆ’1 ,

βˆƒπ›Ώ1 > 0,

(46)

(47)

(52)

βˆ€π‘’ ∈ 𝑋.

βˆ€πœ€ > 0,

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

βˆ—

β€–π‘’β€–π‘βˆ— ≀ 𝐢‖𝑒‖𝑝 ,

π‘βˆ’π‘  𝐴(π‘βˆ’π‘ )/(π‘βˆ’π‘ ) πœ‡(π‘βˆ’π‘)/(π‘βˆ’π‘ ) . 𝑝 (𝑁 βˆ’ 𝑠) 𝑠 (51)

Hence, the proof of the lemma is completed by taking 𝑒0 = Vπœ€ .

By Lemma 10, there exists 𝑒0 ∈ π‘Š0 (Ξ©) with 𝑒0 ≑̸ 0 such that π‘βˆ’π‘  𝐴(π‘βˆ’π‘ )/(π‘βˆ’π‘ ) πœ‡(π‘βˆ’π‘)/(π‘βˆ’π‘ ) . sup 𝐼 (𝑑𝑒0 ) < (58) 𝑝 (𝑁 βˆ’ 𝑠) 𝑠 𝑑β‰₯0 It follows from the nonnegativity of 𝐹(π‘₯, 𝑑) that π‘βˆ— (𝑠)

βˆ— (𝑒0+ ) πœ‡ 1 σ΅„© 󡄩𝑝 𝐼 (𝑑𝑒0 ) = 𝑑𝑝 󡄩󡄩󡄩𝑒0 σ΅„©σ΅„©σ΅„© βˆ’ βˆ— 𝑑𝑝 (𝑠) ∫ 𝑝 𝑝 (𝑠) |π‘₯|𝑠 Ξ©

𝑑π‘₯

βˆ’ ∫ 𝐹 (π‘₯, 𝑑𝑒0 ) 𝑑π‘₯

(59)

Ξ©

π‘βˆ— (𝑠)

βˆ— (𝑒0+ ) πœ‡ 1 σ΅„© 󡄩𝑝 ≀ 𝑑𝑝 󡄩󡄩󡄩𝑒0 σ΅„©σ΅„©σ΅„© βˆ’ βˆ— 𝑑𝑝 (𝑠) ∫ 𝑝 𝑝 (𝑠) |π‘₯|𝑠 Ξ©

𝑑π‘₯.

6

Journal of Applied Mathematics

Therefore, lim𝑑 β†’ +∞ 𝐼(𝑑𝑒0 ) β†’ βˆ’βˆž, so we can choose 𝑑0 > 0 such that σ΅„© σ΅„©σ΅„© 𝐼 (𝑑0 𝑒0 ) ≀ 0. (60) 󡄩󡄩𝑑0 𝑒0 σ΅„©σ΅„©σ΅„© > π‘Ÿ, By virtue of the Mountain Pass Lemma in [16], there is a sequence {𝑒𝑛 } βŠ‚ 𝑋 satisfying 𝐼󸀠 (𝑒𝑛 ) 󳨀→ 0,

𝐼 (𝑒𝑛 ) 󳨀→ 𝑐 β‰₯ 𝛽,

(61)

where 𝑐 = inf max 𝐼 (β„Ž (𝑑)) , β„ŽβˆˆΞ“ π‘‘βˆˆ[0,1]

(62)

Ξ“ = {β„Ž ∈ 𝐢 ([0, 1] , 𝑋) | β„Ž (0) = 0, β„Ž (1) = 𝑑0 𝑒0 } . Note that 0 < 𝛽 ≀ 𝑐 = inf max 𝐼 (β„Ž (𝑑)) ≀ max 𝐼 (𝑑𝑑0 𝑒0 ) ≀ sup 𝐼 (𝑑𝑒0 ) β„ŽβˆˆΞ“ π‘‘βˆˆ[0,1]