Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem

J. Differential Equations 189 (2003) 487–512

Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem$ Yang Haitao Department of Mathematics, Zhejiang University, Hangzhou 310027, China Received September 26 2001; revised May 9 2002

Abstract The following singular elliptic boundary value problem is studied: Du þ lug þ up ¼ 0 u>0

in O; in O;

u¼0

on @O;

where OCRN ðNX3Þ is a bounded domain with smooth boundary @O; 0ogo1oppNþ2 N2; and l > 0 is a real parameter. The existence, multiplicity and asymptotic behavior (as p-1) of solutions of this equation are discussed by combining variational and sub-supersolution methods. r 2002 Elsevier Science (USA). All rights reserved. Keywords: Singular elliptic; Critical exponent; Asymptotic behavior; Sub-super solution

1. Introduction Let O be a bounded domain in RN ; and consider the singular semilinear elliptic problem Du þ f ðx; uÞ ¼ 0

in O;

u>0 u¼0

in O; on @O;

where f ðx; uÞ is singular at u ¼ 0; i.e. f ðx; uÞ-N as u-0: $

This work was supported by the Natural Science Foundation of Zhejiang Province of China. E-mail address: [email protected].

0022-0396/02/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 0 3 9 6 ( 0 2 ) 0 0 0 9 8 - 0

ð1:1Þ

488

Y. Haitao / J. Differential Equations 189 (2003) 487–512

Such problems occur in various branches of mathematical physics. The existence and regularity of solutions for (1.1) have been considered by many authors in recent years (see [1,10–12,14–16,21,22,24]). For instance, Lair and Shaker [14] proved that for f ðx; uÞ ¼ pðxÞf ðuÞ; (1.1) has a unique weak solution uðxÞAH01 ðOÞ if 0ppðxÞAL2 ðOÞ; and f ðuÞ satisfies: (A1): f 0 ðuÞp0: (A2): f ðuÞ > 0; for all u > 0: R (A3): 0e f ðsÞ dsoN; for some e > 0: It is not hard to see that f ðuÞ ¼ ug ð0ogo1Þ satisfies the conditions (A1)–(A3), and it is the nonincreasing condition (A1) that yields the uniqueness of the solution (see also [11,22]). In this paper, we study the following singular elliptic boundary value problem: Du þ lug þ up ¼ 0 in O; u>0 u¼0

in O; on @O;

ð1:2Þ

where OCRN ðNX3Þ is a bounded domain with smooth boundary @O; 0ogo1oppNþ2 N2; and l > 0 is a real parameter. To emphasize the dependence on l or p; problem (1.2) is often referred to as ð1:2Þl or ð1:2Þl;p ; and the subscript l or p is omitted if no confusion arises. A function uðxÞAH01 ðOÞ is called a weak solution to (1.2) if u > 0 in O and Z Z rurj dx ¼ ðlug þ up Þj dx 8jAH01 ðOÞ: ð1:3Þ O

O

It is obvious that the nonlinearity of (1.2) does not satisfy condition (A1) for u sufficiently large. It is natural to ask whether (1.2) has a unique solution or multiple solutions? By combining sub-supersolution and variational methods, we obtain the following results about the structure of the solution set. Theorem 1. Let 0ogo1oppNþ2 N2: Then there exists a constant L > 0 such that: (i) For all lAð0; LÞ; problem ð1:2Þl has at least two weak solutions ul ; vl satisfying ul ovl in O: (ii) For l ¼ L; problem ð1:2Þl has at least one weak solution. (iii) For all lAðL; þNÞ; problem ð1:2Þl has no weak solution.

Remark 1.1. In the case of p ¼ 1 or 0opo1; the uniqueness for problem (1.2) can also be proved by making use of the concavity of the nonlinear term lug þ up (see [9]).

Y. Haitao / J. Differential Equations 189 (2003) 487–512

489

The proof of Theorem 1 is in the spirit of [3–5,8]. In [8], Brezis and Nirenberg proved that for the functional Z 1 jruj2  F ðx; uÞ; IðuÞ ¼ O2 Ru where F ðx; uÞ ¼ 0 f ðx; sÞ ds; local minimizers with respect to the C 1 -topology are also minimizers with respect to the H 1 -topology if jf ðx; uÞjpCð1 þ jujp Þ with ppðN þ 2Þ=ðN  2Þ: Applying this result, Ambrosetti et al. [4] gave the existence of two positive solutions of the following problem: Du þ luq þ up ¼ 0

in O;

u>0 u¼0

in O; on @O;

ð1:4Þ

where O is a bounded domain, and 0oqo1oppNþ2 N2: Some results about multiple solutions were also given by Badiale and Tarantello [5] for some elliptic problems with critical growth and discontinuous nonlinearities. In comparison with problem (1.4) and the problems in [5], the novelty in problem (1.2) lies not only on the nondifferentiability of the corresponding functional, but also on the singularity of (1.2). Since lug þ up - þ N as u-0; the above result of [8] is no longer valid for problem (1.2). There seem to be some difficulties to prove that the first solution obtained by using Perron’s method is a local minimizer of the corresponding functional in H01 ðOÞ; even if a C 1 -minimizer cannot be easily established by applying the maximum-principle (see Section 2). Inspired by Alama [2], we carry out a direct analysis in an H 1 -neighborhood and prove that (1.2) still has a solution which is a local minimizer with respect to the H 1 -topology. Then the existence of a second solution is given by making use of Ekeland’s variational principle. In the case of a subcritical exponent, poðN þ 2Þ=ðN  2Þ; we learn from [10] that there exists ln > 0 such that ð1:2Þl has a solution if lAð0; ln Þ and has no solution if lAðln ; þNÞ: In [24], Sun et al. proved that there exists l * > 0 such that ð1:2Þl has at least two solutions for all lAð0; l * Þ: Their method is to investigate some minimization problems, which is invalid if l is not sufficiently small. Generally speaking, there is a gap between l * and ln ; but Theorem 1 shows that l * ¼ ln : Moreover, when p equals the critical Sobolev exponent, p ¼ ðN þ 2Þ=ðN  2Þ; the problem becomes more delicate because the Sobolev embedding H01 ðOÞ+Lpþ1 ðOÞ is not compact (see [5,7,18–20,25,26]). It should be pointed out that the solutions obtained in Theorem 1 may not be % classical solutions. And for singular equations, a classical solution in C 2 ðOÞ-CðOÞ 1 may not be a weak solution in H0 ðOÞ (see [16] for detail). But under the conditions 0ogoN1 and 1opoðN þ 2Þ=ðN  2Þ; u is a classical solution of (1.2) if and only if u is a weak solution. Based on this development and the methods in [17], we study the

490

Y. Haitao / J. Differential Equations 189 (2003) 487–512

asymptotic behavior of the solutions of ð1:2Þl;p as p-1: Let ul;p ; vl;p denote the solutions ul ; vl of ð1:2Þl;p ; and Lp denote the constant L in Theorem 1. 1 Theorem 2. Let 0ogoN1 and 1opoNþ2 N2: Then uAH0 ðOÞ is a weak solution of (1.2) if % is a classical solution of (1.2). And if s1 > 1; here s1 and only if uAC 2 ðOÞ-C 1þa ðOÞ denotes the principal eigenvalue of D with zero Direchlet condition, then (i) Lp - þ N as p-1: (ii) For any l > 0;

jul;p  u0 jC 1 ðOÞ % -0 as p-1; where u0 denotes the unique solution of the following equation (the existence and uniqueness of u0 can be found in [9,14]) Du þ u þ lug ¼ 0 in O; u>0 in O; u¼0

ð1:5Þ

on @O:

(iii) For any l > 0; jvl;p jLN ðOÞ - þ N

as p-1: cn

Moreover, there exists cn > 0 such that jvl;p jLN ðOÞ Xep1 as p-1: The paper is organized as follows. Section 2 contains the proof of Theorem 1, and the proof of Theorem 2 is given in Section 3. Throughout this paper, we make use of the following notation: Lp ðOÞ; 1pppN; denotes Lebesgue space; the norm in Lp ðOÞ is denoted by j jp ; the norm in H01 ðOÞ is denoted by jj jj; C; C0 ; C1 ; C2 ; y denote (possibly different from line to line) positive constants; supp j ¼ fxjjðxÞa0g; uþ ðxÞ ¼ maxf0; uðxÞg; u ðxÞ ¼ maxf0; uðxÞg:

2. Existence of multiple positive solutions In this section, the proof of Theorem 1 is given. We always assume that 0ogo1 and

p ¼ ðN þ 2Þ=ðN  2Þ:

Because of no lack of compactness, the proof is simple in the case of poðN þ 1Þ=ðN  1Þ: We omit it.

Y. Haitao / J. Differential Equations 189 (2003) 487–512

491

At first, we give the definition of weak super-solution and sub-solution of (1.2). By definition uAH01 ðOÞ is a weak sub-solution to (1.2) if u > 0 in O and Z Z rurjp ðlug þ up Þj 8jAH01 ðOÞ; jX0: O

O

Similarly uAH01 ðOÞ is a weak super-solution to (1.2) if in the above the reverse inequalities hold. Let us define L ¼ supfl > 0jð1:2Þl has a weak solutiong: Lemma 2.1. 0oLo þ N: Proof. Consider the functional 1 l Il ðuÞ ¼ jjujj2  2 1g

Z

juj1g 

O

1 jujpþ1 ; p þ 1 pþ1

uAH01 ðOÞ:

By using Ho¨lder’s and Sobolev’s inequalities, we have Il ðuÞX12 jjujj2  lCjjujj1g  Cjjujjpþ1 ; Moreover, there exist r0 > 0; d0 > 0 such that 8 < 1jjujj2  1 jujpþ1 X2d0 2 pþ1 pþ1 : 1jjujj2  2

pþ1 1 pþ1jujpþ1 X0

uAH01 ðOÞ:

8uA@Br0 ; 8uABr0 ;

ð2:1Þ

where Br0 ¼ fuAH01 ðOÞ : jjujjpr0 g: Then we can choose l * > 0; small enough such that Il j@Br Xd0 > 0: *

0

ð2:2Þ

Set c * ¼ inf Il : Br 0

*

Since 0o1  go1; for every vc0; Il ðtvÞo0 provided t > 0 is sufficiently small. This * implies that c * o0: Let fuj gCBr0 be a minimizing sequence for c * : Then there exists a subsequence of fuj g (still denoted by fuj g) such that uj -u * weakly in H01 ðOÞ; strongly in Lq ðOÞ for 2N 2pqoN2 ; and pointwise a.e. in O: Since Il ðjujÞ ¼ Il ðuÞ; we may assume uj X0: By

Y. Haitao / J. Differential Equations 189 (2003) 487–512

492

using Ho¨lder’s inequality, we get that as j-N; Z Z Z 1g u1g p u þ juj  u * j1g j * O O O Z p u1g þ Cjuj  u * j1g 2 * O Z ¼ u1g þ oð1Þ: *

O

Similarly, Z O

u1g p *

Z O

u1g j

þ

Z

1g

O

juj  u * j

¼

Z O

u1g þ oð1Þ: j

Then Z O

u1g j

¼

Z O

u1g þ oð1Þ: *

ð2:3Þ

By the Brezis–Lieb lemma [6], pþ1 pþ1 juj jpþ1 pþ1 ¼ ju * jpþ1 þ juj  u * jpþ1 þ oð1Þ:

ð2:4Þ

jjuj jj2 ¼ jju * jj2 þ jjuj  u * jj2 þ oð1Þ:

ð2:5Þ

Moreover,

It follows from (2.2) and c * o0 that jjuj jjpr0  e0 for some constant e0 > 0 independent of j: Then, by (2.5), u * ABr0 and uj  u * ABr0 for j sufficiently large. Therefore (2.1) implies 1 1 jjuj  u * jj2  juj  u * jpþ1 pþ1 X0: 2 pþ1 Using this inequality and (2.3)–(2.5), we infer that as j-N; c * ¼ Il ðuj Þ þ oð1Þ *

1 1 juj  u * jpþ1 ¼ Il ðu * Þ þ jjuj  u * jj2  pþ1 þ oð1Þ * 2 pþ1 X Il ðu * Þ þ oð1ÞXc * þ oð1Þ; *

i.e. Il ðu * Þ ¼ c * o0: Hence 0pu * c0 is a local minimizer of Il in H01 ðOÞ: * * Now, by the same arguments as in [14,24], one can prove that u * is a weak solution of ð1:2Þl : For the convenience of the reader, we sketch the main steps here. *

Y. Haitao / J. Differential Equations 189 (2003) 487–512

493

For any fAH01 ðOÞ; fX0; lim inf þ t-0

Il ðu * þ tfÞ  Il ðu * Þ * * X0: t

ð2:6Þ

From this, one derives that Du * X0 in the weak sense. By the strong maximum principle, u * > 0 in O: Moreover, using Fatou’s lemma in (2.6) yields Z Z ru * rf dxX ðlug þ up Þf dx 8fAH01 ðOÞ; fX0: ð2:7Þ O

*

O

*

For any jAH01 ðOÞ and e > 0; taking f ¼ ðu * þ ejÞþ in (2.7), dividing it by e and letting e-0; one has Z Z ru * rj dxX ðlug þ up Þj dx 8jAH01 ðOÞ: ð2:8Þ O

O

*

*

Replacing j by j in (2.8), one gets the conclusion. Thus LXl * > 0: It remains to show that Lo þ N: Let s1 denote the principal eigenvalue of D with zero Direchlet condition, and e1 denote the corresponding eigenfunction. Multiplying ð1:2Þl by e1 and integrating over O we have Z Z s1 ue1 ¼ ðlug þ up Þe1 : O

O

Let Ln be a constant such that Ln tg þ tp > 2s1 t 8t > 0: This implies that loLn ; and then LoLn o þ N: The proof of this lemma is completed. & In the following two lemmas, we give the existence of solutions of (1.2) by using Perron’s method in a variational guise [23]. 1 Lemma 2.2. Suppose uAH01 ðOÞ is a weak sub-solution while uAH % 0 ðOÞ is a weak super% solution to problem (1.2) such that upu% a.e. in O: Then there exists a weak solution % uAH01 ðOÞ of (1.2) satisfying the condition upupu% a.e. in O: %

Proof. We follow the same method of Perron’s except that here the functional Il is nondifferentiable in H01 ðOÞ: Set M ¼ fuAH01 ðOÞjupupu; % a:e: in Og: % Obviously M is closed and convex, and Il is weakly lower semicontinuous on M: In fact, if um AM and um ,u weakly in H01 ðOÞ; we may assume that um -u R R 1þp 1g pointwise a.e. in O (along a subsequence). Since O juj o þ N and O juj oþ % % N; it follows from Lebesgue’s dominated convergence theorem that R R R R 1þp - O u1þp and O jum j1g - O u1g : Then lim inf n-N Il ðum ÞXIl ðuÞ: O jum j

494

Y. Haitao / J. Differential Equations 189 (2003) 487–512

So by Lemma 1.2 of [23], there exists a relative minimizer u of Il on M: To show that u solves problem (1.2), for jAH01 ðOÞ and e > 0; let ve ¼ u þ ej  je þ je AM with þ je ¼ ðu þ ej  uÞ % X0

and je ¼ ðu þ ej  uÞ X0: % As 0oto1; u þ tðve  uÞAM: Then Il ðu þ tðve  uÞÞ  Il ðuÞ 0p lim t-0 t Z Z ¼ rurðve  uÞ  l lim ðu þ ytðve  uÞÞg ðve  uÞ t-0 O O Z  up ðve  uÞ;

ð2:9Þ

O

0oyo1: Since u is a weak sub-solution of (1.2) and jve  ujAH01 ðOÞ; Rwhere % g g g O u jve  ujo þ N: Moreover, jðu þ ytðve  uÞÞ ðve  uÞjpu jve  uj: Then, by % % Lebesgue’s theorem again, we have Z Z lim ðu þ ytðve  uÞÞg ðve  uÞ ¼ ug ðve  uÞ: ð2:10Þ t-0

O

O

Therefore, by (2.9) and (2.10), Z Z Z 0p rurðve  uÞ  l ug ðve  uÞ  up ðve  uÞ; O

O

O

i.e. Z

1 rurj  lug j  up jX ðE e  Ee Þ; e O

where Ee ¼

Z

rurje  lug je  up je

O

and Ee ¼

Z

rurje  lug je  up je : O

ð2:11Þ

Y. Haitao / J. Differential Equations 189 (2003) 487–512

495

Furthermore, Z  Z 1 e 1 e e g p e E ¼ rðu  uÞrj þ rurj  ðlu þ u Þj % % e e O O Z Z 1 2 jrðu  uÞj rðu  uÞrj X % þ % e Oe Oe Z 1 þ ðlu% g þ u% p  lug  up Þje e Oe Z Z X rðu  uÞrj  l ju% g  ug j jjj % Oe

¼ oð1Þ

Oe

ð2:12Þ

as e-0;

> uðxÞg and measðOe Þ-0 as e-0: Similarly, where Oe ¼ fxAOjuðxÞ þ ejðxÞXuðxÞ % 1 Ee poð1Þ: e

ð2:13Þ

It follows from (2.11) to (2.13) that Z rurj  lug j  up jXoð1Þ as e-0: O

Reversing the sign of j and letting e-0; we obtain Z rurj  lug j  up j ¼ 0 8jAH01 ðOÞ: O

The proof of this lemma is completed. & Lemma 2.3. For lAð0; L ; ð1:2Þl has a weak solution ul AH01 ðOÞ: Proof. Fix lAð0; LÞ; and consider the problem Du þ lug ¼ 0 in O; u>0 in O; u¼0

ð2:14Þ

on @O:

By the results in [11,14,22], (2.14) has a unique weak solution wl : Moreover, since 0ogo1; we have Jl ðwl Þ ¼

inf

uAH01 ðOÞ

Jl ðuÞo0;

where Jl is the corresponding functional for problem (2.14), i.e. Z 1 l Jl ðuÞ ¼ jjujj2  juj1g ; uAH01 ðOÞ: 2 1g O

496

Y. Haitao / J. Differential Equations 189 (2003) 487–512

By the definition of L; there exists l0 Aðl; LÞ such that ð1:2Þl0 has a solution ul0 : Note that wl is a sub-solution, and ul0 is a super-solution of ð1:2Þl : Claim. wl pul% a.e. in O for any l% A½l; LÞ: In fact, let yðtÞ be a smooth nondecreasing function such that yðtÞ  1 for tX1 and y  0 for tp0: Set ye ðtÞ ¼ yðetÞ: Using ye ðwl  ul% Þ as a test function in ð2:1Þl% and (2.14), we infer Z 0X  jrðul%  wl Þj2 y0e ðwl  ul% Þ O Z p ¼ ðl% ug  lwg l þ ul% Þye ðwl  ul% Þ l% O Z X l ðug  wg l Þye ðwl  ul% Þ: l% O

As e-0; we are led to Z wl >ul%

ðug  wg l Þp0: l%

This implies that measðfxAOjwl ðxÞ > ul% ðxÞgÞ ¼ 0: The claim is proved. Applying Lemma 2.2 with u% ¼ ul0 and u ¼ wl shows that ð1:2Þl for lAð0; LÞ has a % solution ul satisfying Il ðul ÞpIl ðwl ÞpJl ðwl Þo0: For l ¼ L; let ln Að0; LÞ be an increasing sequence such that ln -L; and uln be the solution of ð1:2Þln obtained above. Then Z Z 1 ln 1 Iln ðuln Þ ¼ jjuln jj2  juln j1g  jul jpþ1 o0 ð2:15Þ 2 pþ1 O n 1g O and jjuln jj2  ln

Z O

juln j1g 

Z

juln jpþ1 ¼ 0:

ð2:16Þ

O

It follows from (2.15) and (2.16) that fuln g is bounded in H01 ðOÞ: Thus, there exists uL AH01 ðOÞ such that uln -uL weakly in H01 ðOÞ (along a subsequence) and pointwise a.e. in O: By the claim, uL Xwl1 > 0 a.e. in O: Letting n-N in ð1:2Þln in the weak sense, and by using Lebesgue’s theorem as in Lemma 2.2, we get that uL is a weak solution of ð1:2ÞL : The proof of this lemma is completed. & In order to give the existence of a second solution of ð1:2Þl with lAð0; LÞ; we need to show that the solution ul obtained in Lemmas 2.2 and 2.3 is a local minimizer of Il in H01 ðOÞ: Note that by Lemma 2.1 this is true for l sufficiently small. But for l not close to zero, ul is only the constrained minimizer of Il on M: Because of the

Y. Haitao / J. Differential Equations 189 (2003) 487–512

497

singularity of problem (1.2), it is not obvious that ul is a local C 1 -minimizer if one uses the maximum principle. A direct analysis in an H 1 -neighborhood is required (see [2]). We need the following lemma. Lemma 2.4 (Brezis and Nirenberg [8]). Let O be a bounded domain in RN with smooth boundary @O: Let uAL1loc ðOÞ and assume that, for some kX0; u satisfies, in the sense of distributions, Du þ kuX0 in O; uX0 in O: Then either u  0; or there exists e > 0 such that uðxÞXe distðx; @OÞ;

xAO:

Proof. See [8, Theorem 3]. Lemma 2.5. Let u% be the super-solution, u be the sub-solution and ul be the solution of ð1:2Þl with lAð0; LÞ obtained in Lemma% 2.3. Then ul is a local minimizer for Il in H01 ðOÞ: Proof. If the conclusion is not true, there exists a sequence fun gCH01 ðOÞ such that un -ul strongly in H01 ðOÞ and Il ðun ÞoIl ðul Þ: Set vn ¼ maxfu; minfun ; ugg; % % þ w% n ¼ ðun  uÞ % ;

S% n ¼ supp w% n ;

wn ¼ ðun  uÞ ; % % S n ¼ supp wn : % %

Then un ¼ vn  wn þ w% n ; vn AM ¼ fuAH01 ðOÞjupupug % and % % Il ðun Þ ¼ Il ðvn Þ þ An þ Bn ; where An ¼

Z

1 2 ðjrun j2  jruj % Þ S% n 2  Z  l 1 l 1 jun j1g þ jun jpþ1  u% 1g  u% pþ1  1þp 1g 1þp S% n 1  g

Y. Haitao / J. Differential Equations 189 (2003) 487–512

498

and Bn ¼

Z

1 ðjrun j2  jruj2 Þ % Sn 2 % Z   l 1 l 1 jun j1g þ jun jpþ1  u1g  upþ1 :  1þp 1g% 1þp% Sn 1  g %

Recalling that Il ðul Þ ¼ inf uAM Il ðuÞ; we have Il ðun ÞXIl ðul Þ þ An þ Bn :

ð2:17Þ

To estimate An and Bn ; we need to show that as n-N; measðS% n Þ-0;

measðSn Þ-0: %

ð2:18Þ

In fact, for any e > 0; there exists a constant d > 0 such that measðO\Od Þo2e ; where Od ¼ fxAOjdistðx; @OÞ > dg: By Lemma 2.4, uðxÞXC1 distðx; @OÞXC21 d > 0 % for xAOd : Then 2

Dðu%  ul ÞX lðu% g  ug l Þ ¼  lgðyu% þ ð1  yÞul Þg1 ðu%  ul Þ ð0oyo1Þ  C1 d g1 X  lg ðu%  ul Þ 2

in Od : 2

Applying Lemma 2.4 again implies that uðxÞ  ul ðxÞXC2 distðx; @Od ÞXC22 d > 0 for % 2

xAOd : Since un -ul in H01 ðOÞ; it follows that for n sufficiently large, measðS% n Þp measðO\Od Þ þ measðS% n -Od Þ Z e 4 o þ 2 2 ðun  ul Þ2 2 C2 d O o e;

i.e. measðS% n Þ-0 as n-N: Similarly, measðS n Þ-0 as n-N: % Thus, by (2.18 ), we have Z 2 jjw% n jj2 ¼ jrðun  uÞj % S% n Z 2 p 2jjun  ul jj2 þ 2 jrðul  uÞj % -0 as n-N: S% n

ð2:19Þ

Y. Haitao / J. Differential Equations 189 (2003) 487–512

499

Since u% is a super-solution of (1.2), we infer Z 1 2 ðjrðu% þ w% n Þj2  jruj An ¼ % Þ 2 % Sn  Z  l 1 l 1 1g pþ1 1g pþ1 ðu% þ w% n Þ þ ðu% þ w% n Þ u% u%    1þp 1g 1þp S% n 1  g Z 1 ¼ jjw% n jj2 þ rur % w% n 2 O  Z  l 1 l 1 1g pþ1 1g pþ1 ðu% þ w% n Þ þ ðu% þ w% n Þ u% u%    1þp 1g 1þp S% n 1  g Z 1 ðlu% g þ u% p Þw% n X jjw% n jj2 þ 2 % Sn Z  lðu% þ yw% n Þg w% n þ ðu% þ yw% n Þp w% n ð0oyo1Þ % Sn Z 1 X jjw% n jj2 þ ½u% p  ðu% þ yw% n Þp w% n 2 S% n Z 1 X jjw% n jj2  C ðu% p1 þ w% p1 % 2n : n Þw 2 S% n It follows from (2.18), (2.19), Ho¨lder’s and Sobolev’s inequalities that for n sufficiently large, Z p1 pþ1 1 2 pþ1 An X jjw% n jj  C u% jjw% n jj2  Cjjw% n jjpþ1 2 S% n 1 ¼ jjw% n jj2  oð1Þjjw% n jj2 X0: 2 Similarly, Bn X0: Therefore, by (2.17), Il ðun ÞXIl ðul Þ: This is a contradiction to our assumption Il ðun ÞoIl ðul Þ: The proof of this lemma is completed. & Remark 2.1. It is clear from the proof that in Lemma 2.5 one can substitute ug with any decreasing function (see also [2, Proposition 5.2]). Now, we are ready to find a second solution of ð1:2Þl by making use of Ekeland’s variational principle on the following set: T ¼ fuAH01 ðOÞjuXul ; a:e: in Og; where ul is given in Lemma 2.3. By Lemma 2.5, there exists 0ol0 pjjul jj such that Il ðuÞXIl ðul Þ; 8u with jju  ul jjp! 0 : Then one of the following cases holds: ðP1 Þ InffIl ðuÞjuAT; jju  ul jj ¼ lg ¼ Il ðul Þ 8lAð0; l0 Þ: ðP2 Þ There exists l1 Að0; l0 Þ such that inffIl ðuÞjuAT; jju  ul jj ¼ l1 g > Il ðul Þ: Following the arguments in [5], we treat separately cases ðP1 Þ and ðP2 Þ:

500

Y. Haitao / J. Differential Equations 189 (2003) 487–512

Lemma 2.6. Assume lAð0; LÞ and ðP1 Þ: Then for any lAð0; l0 Þ; there exists a solution vl of ð1:2Þl such that 0oul ovl in O and jjul  vl jj ¼ l: Proof. Fix lAð0; l0 Þ; and let r > 0 such that l  r > 0 and l þ rol0 : Set R ¼ fuATj0ol  rpjju  ul jjpl þ rg: Obviously, R is a closed set in H01 ðOÞ; and by ðP1 Þ; inf R Il ¼ Il ðul Þ: For any minimizing sequence fun g; by Ekeland’s variational principle, there exists a sequence fvn gCR satisfying 8 1 > < Il ðvn ÞpIl ðun ÞpIl ðul Þ þ n; jjun  vn jjp1n; ð2:20Þ > : 1 Il ðvn ÞpIl ðvÞ þ njjv  vn jj 8vAR: For wAT; as e > 0 is sufficiently small, vn þ eðw  vn ÞAR: Hence by (2.20) Il ðvn þ eðw  vn ÞÞ  Il ðvn Þ 1 X  jjw  vn jj: e n Letting e-0; we get 1  jjw  vn jj n Z Z p rvn rðw  vn Þ  vpn ðw  vn Þ O

O

Z l ðvn þ eðw  vn ÞÞ1g  v1g n lim inf  1  g e-0 e O Z Z rvn rðw  vn Þ  vpn ðw  vn Þ ¼ O O Z  l lim inf ðvn þ eyðw  vn ÞÞg ðw  vn Þ; e-0

O

where 0oyo1: Similar to (2.10), we have Z Z g lim inf ðvn þ eyðw  vn ÞÞ ðw  vn Þ ¼ vg n ðw  vn Þ: e-0

O

O

Then 1  jjw  vn jjp n

Z O

rvn rðw  vn Þ 

Z O

vpn ðw

 vn Þ  l

Z O

vg n ðw  vn Þ

ð2:21Þ

for all wAT: Since fvn g is bounded in H01 ðOÞ; we may assume (without loss of generality) that vn -vl AT weakly in H01 ðOÞ and pointwise a.e. in O: At first, we show

Y. Haitao / J. Differential Equations 189 (2003) 487–512

501

that vl is a solution of ð1:2Þl as in [5]. For jAH01 ðOÞ and e > 0; set fn;e ¼ ðvn þ ej  ul Þ AH01 ðOÞ: Obviously vn þ ej þ fn;e AT: Replacing w in (2.21) by this function, we get Z Z 1  jjej þ fn;e jjp rvn rðej þ fn;e Þ  l vg n ðej þ fn;e Þ n O O Z  vpn ðej þ fn;e Þ: ð2:22Þ O

g Since jfn;e jpul þ ejjj; jvpn fn;e jpðul þ ejjjÞp and jvg n ðej þ fn;e Þjpul ðul þ 2ejjjÞ; Lebesgue’s theorem may be employed to show that Z Z p vn ðej þ fn;e Þ- vpl ðej þ fe Þ as n-N ð2:23Þ O

O

and Z O

vg n ðej

þ fn;e Þ-

Z O

vg l ðej þ fe Þ as n-N;

ð2:24Þ

where fe ¼ ðvl þ ej  ul Þ : Since measðfxAOjvn þ ejoul pvl þ ejgÞ-0 as n-N; one has (see [5]) Z rvn rfn;e O Z Z p rvn rðul  vl  ejÞ þ rvl rðvl  vn Þ þ oð1Þ u >v þej ul >vn þej Zl l rvl rfe þ oð1Þ as n-N: ð2:25Þ ¼ O

Using (2.23)–(2.25) in (2.22) and letting n-N; we infer Z p rvl rj  lvg l j  vl j O Z 1 p g rvl rfe  lvl fe  vl fe X e O Z 1 p p g g ¼ rðvl  ul Þrfe þ lðvl  ul Þfe þ ðvl  ul Þfe e O Z 1 g g X rðvl  ul Þrðul  vl  ejÞ þ lðvl  ul Þðul  vl  ejÞ e ul >vl þej  Z Z 1 g g X e rðvl  ul Þrj þ le ðvl  ul Þj e ul >vl þej ul >vl þej ¼ oð1Þ

as e-0:

ð2:26Þ

502

Y. Haitao / J. Differential Equations 189 (2003) 487–512

The last equality holds because measðfxAOjvl ðxÞ > ul ðxÞ > vl ðxÞ þ ejðxÞgÞ-0 as e-0: Letting e-0 and reversing the sign of j; we have Z p 1 rvl rj  lvg l j  vl j ¼ 0 8jAH0 ðOÞ; O

i.e. vl is a weak solution of ð1:2Þl : To prove that vl cul ; it suffices to show that vn -vl strongly in H01 ðOÞ: Similar to (2.3)–(2.5), as n-N; one obtains jjvn jj2 ¼ jjvn  vl jj2 þ jjvl jj2 þ oð1Þ; pþ1 pþ1 jvn jpþ1 pþ1 ¼ jvn  vl jpþ1 þ jvl jpþ1 þ oð1Þ;

Z O

v1g n

¼

Z O

v1g þ oð1Þ l

and Z

jvn  vl j1g ¼ oð1Þ:

O

Taking w ¼ vl in (2.21), we get Z Z Z Z 1g jrðvn  vl Þj2 þ l vg v p l v þ vpn ðvn  vl Þ þ oð1Þ n l n O O O O Z p l v1g þ jvn  vl jpþ1 pþ1 þ oð1Þ: l

ð2:27Þ

O

By Lebesgue’s theorem again,

R 1g g O vn v l - O vl

R

as n-N: Then

jjvn  vl jj2 pjvn  vl jpþ1 pþ1 þ oð1Þ

as n-N:

ð2:28Þ

On the other hand, taking w ¼ 2vn in (2.21) yields Z jjvn jj2  jvn jpþ1  l v1g n Xoð1Þ as n-N: pþ1

ð2:29Þ

Recalling that vl is a solution of (1.2), we have Z pþ1 2 jjvl jj  jvl jpþ1  l v1g ¼ 0: l

ð2:30Þ

O

O

It follows from (2.29) and (2.30) that jjvn  vl jj2 Xjvn  vl jpþ1 pþ1 þ oð1Þ

as n-N:

ð2:31Þ

Y. Haitao / J. Differential Equations 189 (2003) 487–512

503

Thus, by (2.28) and (2.31), jjvn  vl jj2 ¼ jvn  vl jpþ1 pþ1 þ oð1Þ as n-N:

ð2:32Þ

Without loss of generality, one may assume Il ðul ÞpIl ðvl Þ: So by (2.20), 1 Il ðvn  vl ÞpIl ðvn Þ  Il ðvl Þ þ oð1ÞpIl ðul Þ  Il ðvl Þ þ þ oð1Þpoð1Þ n as n-N: Then 1 1 jjvn  vl jj2  jvn  vl jpþ1 pþ1 poð1Þ as n-N: 2 pþ1

ð2:33Þ

Therefore, by (2.32) and (2.33), vn -vl strongly in H01 ðOÞ: Moreover, since jjun  ul jj ¼ l and jjun  vn jjp1n; jjvl  ul jj ¼ l: This implies that vl cul : By using Lemma % 2.4 in any B; where BCO; we get that ul ovl in O: The proof of this lemma is completed. & Lemma 2.7. Assume lAð0; LÞ and ðP2 Þ: Then there exists a solution vl of ð1:2Þl such that 0oul ovl in O: Proof. Define the complete metric space G ¼ fZACð½0; 1 ; TÞjZð0Þ ¼ ul ; jjZð1Þ  ul jj > l1 ; Il ðZð1ÞÞoIl ðul Þg with distance dðZ0 ; ZÞ ¼ max jjZ0 ðtÞ  ZðtÞjj tA½0;1

8Z0 ; ZAG:

Set g0 ¼ inf max Il ðZðtÞÞ: ZAG tA½0;1

At first, we need to show that Ga| and to estimate the minimax level g0 for which compactness can be established. As usual (e.g. [5,7]), we consider Ue ðxÞ ¼

CN eðN2Þ=2 ðe2 þ jx  yj2 ÞðN2Þ=2

jðxÞ;

where CN is a normalization constant, yAO and jAC0N ðOÞ is a fixed function such that jðxÞ ¼ 1 for x in some neighborhood of y:

Y. Haitao / J. Differential Equations 189 (2003) 487–512

504

Claim. There exist e0 > 0 and R0 X1 such that ( Il ðul þ RUe ÞoIl ðul Þ Il ðul þ tR0 Ue ÞoIl ðul Þ þ

1 N

S

8eAð0; e0 Þ 8RXR0 ;

N=2

8tA½0; 1 8eAð0; e0 Þ;

where S is the best Sobolev constant. Indeed, by Lemma 3.3 of [5], for a suitable bAð0; p þ 1Þ; one gets Z 1 1 pþ1 pþ1 t R A  Rp tp Uep ul dx Il ðul þ tRUe Þ ¼ Il ðul Þ þ R2 t2 B  2 pþ1 O þ De þ Rb oðeðN2Þ=2 Þ;

ð2:34Þ

where B¼

Z

2

jrU1 j dx;

A¼

Z

RN

RN

1 ð1 þ jxj2 ÞN

dx

and De ¼

Z  O

lug l tRUe þ

 1 1 1g lu1g lðu  þ tRU Þ dx: l e 1g l 1g

We only need to estimate De : Let t be a constant such that 0oto14; then as e-0;   Z 1 1 g 1g 1g lu  lðul þ tRUe Þ De ¼ lul tRUe þ dx 1g l 1g jxyjpet   Z 1 1 1g 1g lu lðu lug tRU þ  þ tRU Þ dx þ e l e l 1g l 1g jxyj>et Z Z Ue dx þ lð1  gÞgðul þ tRyUe Þg1 ðtRUe Þ2 dx p CtR jxyjpet

jxyj>et

ð0oyo1Þ p CtR

Z

CN eðN2Þ=2

dx þ Ct2 R2 ðN2Þ=2

ðe2 þ jx  yj2 Þ Z et Z p CtReðN2Þ=2 r dr þ Ct2 R2 jxyjpet

0

Z jxyj>et

Ue2 dx

eN2 jxyj>et

ðe2 þ jx  yj2 ÞN2

dx

p CtReðN2Þ=2þ2t þ Ct2 R2 jOjeN22tðN2Þ ¼ CtRoðeðN2Þ=2 Þ þ Ct2 R2 oðeðN2Þ=2 Þ:

ð2:35Þ

By (2.34) and (2.35), the claim holds if one follows the arguments of Lemma 3.3 in [5] word for word (see also [25]).

Y. Haitao / J. Differential Equations 189 (2003) 487–512

505

Thus, ZðtÞ ¼ ul þ tR0 Ue ; tA½0; 1 belongs to G: Then Ga| and Il ðul Þog0 oIl ðul Þ þ

1 N=2 S : N

ð2:36Þ

Now, applying Ekeland’s variational principle on G and arguing exactly as in the proof of Lemma 3.5 in [5], one can prove that there exists vk AT such that 8 > < RIl ðvk Þ-g0 as k-N; R p R g O rvk rðw  vk Þ  O vk ðw  vk Þ  l O vk ðw  vk Þ > : X  Ck ð1 þ jjwjjÞ 8wAT:

ð2:37Þ

Let w ¼ 2vk in (2.37), we have g0 þ oð1Þ Z Z 1 l 1 jvk j1g  jvk jpþ1 ¼ jjvk jj2  2 1g O pþ1 O   Z 1 1 1 1 C  ð1 þ jj2vk jjÞ: X  jvk j1g  jjvk jj2  l 2 pþ1 1g pþ1 O kðp þ 1Þ Then fvk g is bounded in H01 ðOÞ: Similar to Lemma 2.6, it can be derived from (2.37) that vk converges to a solution vl of ð1:2Þl weakly in H01 ðOÞ and jjvk  vl jj2  jvk  vl jpþ1 pþ1 ¼ oð1Þ

as k-N:

ð2:38Þ

By (2.36), for a suitable e00 > 0; one finds 1 1 jjvk  vl jj2  jvk  vl jpþ1 pþ1 ¼ Il ðvk Þ  Il ðvl Þ þ oð1Þ 2 pþ1 p g0  Il ðul Þ þ oð1Þ o

1 N=2 S  e00 : N

ð2:39Þ

Then, by (2.38) and (2.39), it can be proved (see [5]) that vk -vl strongly in H01 ðOÞ; and then Il ðvl Þ ¼ g0 > Il ðul Þ: Therefore ul cvl in O: By using Lemma 2.4, it follows that vl > ul in O: The proof of this lemma is completed. & Proof of Theorem 1. Theorem 1 now follows from Lemmas 2.3, 2.6, 2.7 and the definition of L:

506

Y. Haitao / J. Differential Equations 189 (2003) 487–512

3. Asymptotic behavior of solutions as p-1 In this section, the proof of Theorem 2 is given. At first, we prove the following regularity result which is based on Lemma 2.4 and Moser’s iteration technique (see [13]). 1 Lemma 3.1. Let 0ogoN1 and 1opoNþ2 N2: Then uAH0 ðOÞ is a weak solution of (1.2) if % is a classical solution of (1.2). Moreover, and only if uAC 2 ðOÞ-C 1þa ðOÞ m

p jujC 1þa ðOÞ % pCð1 þ jjujj Þ;

where C and m are independent of p; 0oao1: Proof. Let u be a classical solution of (1.2). To prove that u is a weak solution of (1.2), it suffices to show that for all jAH01 ðOÞ; Z

ug jo þ N:

O

In fact, by Lemma 2.4, uðxÞXC distðx; @OÞ for xAO: Then by using Ho¨lder’s inequality and the assumption 0ogoN1 ; we have Z

Z

g

u jp O

u

2Ng Nþ2

Nþ2 2N

O

pC

Z

jjjL2n 2Ng Nþ2

ðdistðx; @OÞÞ

Nþ2 2N

O

o þ N;

jjjL2n ð3:1Þ

2N : where 2n ¼ N2 On the other hand, if u is a weak solution of (1.2), then by Lemma 2.4 and the claim in Lemma 2.3, there exists C > 0 independent of p such that uðxÞXwl ðxÞXC distðx; @OÞ for xAO: It follows that there exists q0 > N satisfying

Z O

ugq0 p

Z O

0 wgq pCo þ N: l

2n

By Sobolev’s embedding theorem, up AL p ðOÞ: Without loss of generality, we may n assume that 2p oN: Applying the Calde´ron–Zygmund inequality [23, Theorem B.2],

Y. Haitao / J. Differential Equations 189 (2003) 487–512

507

we get ! juj

2n p C 2; W p

pC

g

lju j

p

þ ju j

2n Lp

n ljwg l j 2 Lp

p

2n Lp

þ ju j

  p C 1 þ jujpL2n :

!

2n Lp

ð3:2Þ

Thus Sobolev’s inequality implies that jujLt1 pCð1 þ jujpL2n Þ;

where t1 ¼

2n N p 2n N2 p

¼ d0 2n ; d0 ¼

1 2 2n p N

ð3:3Þ

> 1:

Replacing 2n by t1 in (3.2) and (3.3), we have     2 jujLt2 pC 1 þ jujpLt1 pC 1 þ jujpL2n ;

where t2 ¼

t N p1 t N2 p1

> d0 t1 ¼ d20 2n :

Replacing t1 by t2 and repeating the process so on, we finally obtain a positive integer m such that

n dm 02 p

> N and % Cjuj jujC 1þa ðOÞp

2;minfq0 ;

W g

n dm 02 p g

p Cðlju jLq0 þ juj

nÞ dm 02 p L

m

p Cð1 þ jujpL2n Þ m

p Cð1 þ jjujjp Þ; where C and m are independent of p: At last, standard regularity theory can be used to show that uAC 2 ðOÞ: The proof of this lemma is completed. & Remark 3.1. From the proof of Lemma 3.1, it is not hard to see that if 0ogoN2 ; % uAC 2 ðOÞ-C a ðOÞ: Lemma 3.2. Let 0ogoN1 and s1 > 1: Then Lp - þ N as p-1; where s1 denotes the principal eigenvalue of D with zero Direchlet condition, and Lp is as in Theorem 1.

508

Y. Haitao / J. Differential Equations 189 (2003) 487–512

Proof. For any l > 0; let d > 0 such that 1 þ dos1 and ud be the unique solution of the equation Du þ ð1 þ dÞu þ lug ¼ 0

in O;

u>0 u¼0

in O; on @O:

ð3:4Þ

Since 0ogoN1 ; the same argument as in Lemma 3.1 can be used to show that % Then ud AC 2 ðOÞ-C 1þa ðOÞ: p p1 Dud X lug d þ ud þ ðð1 þ dÞ  jud jLN ðOÞ Þud p X lug d þ ud

as p-1;

i.e. ud is a supersolution of ð1:2Þl : Let wl be the solution of (2.14), which is a subsolution of ð1:2Þl : By the claim in Lemma 2.3, ud Xwl ; and ð1:2Þl has a solution. The conclusion follows from the definition of Lp : Remark 3.2. If s1 p1; it is not hard to see that Lp -0 as p-1: In fact, multiplying ð1:2Þl by the first eigenfunction j1 of D and integrating over O; we have Z ðlug þ up  s1 uÞj1 ¼ 0: O

But for any l > 0; lug þ up  s1 u > 0 8u > 0; as p-1: Fix l > 0; by Lemma 3.2, lAð0; LÞ as p-1: And ð1:2Þl;p has two solutions ul;p ; vl;p obtained in Lemmas 2.3, 2.6 or 2.7. For simplicity, let up denote ul;p ; and vp denote vl;p in the following. % as Lemma 3.3. Let 0ogoN1 and s1 > 1: Then fup g is bounded in H01 ðOÞ-C 1þa ðOÞ p-1: Proof. By Lemma 2.3, Il ðup Þp0; i.e. Z Z 1 l 1 jjup jj2  jup j1g  jup jpþ1 p0 2 1g O pþ1 O and jjup jj2 ¼ l

Z

jup j1g þ

O

Z

jup jpþ1 : O

Putting together these relations and using Ho¨lder’s inequality, we find 1 1þg

jjup jjpCðp  1Þ

:

Y. Haitao / J. Differential Equations 189 (2003) 487–512

509

By Lemma 3.1, as p-1; jup jC 1þa ðOÞ % pCð1 þ ðp  1Þ

pm 1þg

ÞpCðp  1Þ1 ;

ð3:5Þ

where C may be different from line to line and independent of p: Now, we prove the result by contradiction. Suppose that there exists pi -1 such u that jjupi jj- þ N as i- þ N: Set wi ¼ jjuppi jj: Since fwi g is bounded in H01 ðOÞ; we may i

assume that there exists 0pw0 AH01 ðOÞ such that wi -w0 weakly in H01 ðOÞ and strongly in L2 ðOÞ: Then 1 ¼l ¼

Z

Z

u1g pi

O

O

jjupi jj

þ 2

Z

uppii þ1 O

jjupi jj2

w20 þ Ai þ oð1Þ

as i- þ N;

ð3:6Þ

where  Z  upi þ1  u2   pi pi  jAi j ¼    O jjupi jj2    Z 1þxi  upi ln upi   ¼ ðpi  1Þ  ð1oxi opi Þ  jjupi jj2  O 2

m xi

p Cðpi  1Þpi

j lnCðpi  1Þ1 j þ oð1Þ

¼ oð1Þ as pi -1: Thus 0pw0 c0: Similar to (3.7), we get that for any jAH01 ðOÞ; Z pi up i  up i j ¼ oð1Þ as i- þ N: O jjupi jj By the claim in Lemma 2.3, we have Z  Z   ug wg pi l  p

j

jjj ¼ oð1Þ:  jju jj  pi O O jjupi jj

ð3:7Þ

ð3:8Þ

ð3:9Þ

Dividing ð1:3Þl;pi by jjupi jj and integrating over O; we infer from (3.8) and (3.9) that as i- þ N; Z Z Z ug uppii pi

jþ j rwi rj ¼ l O O jjupi jj O jjupi jj Z ¼ w0 j þ oð1Þ: ð3:10Þ O

Y. Haitao / J. Differential Equations 189 (2003) 487–512

510

R R Let i- þ N; and obtain O rw0 rj ¼ O w0 j; which contradicts the assumption % s1 > 1: Therefore fup g is bounded in H01 ðOÞ-C 1þa ðOÞ: & Proof of Theorem 2. In view of Lemmas 3.1 and 3.2, it remains to prove (ii) and (iii). % To prove (ii), for any pi -1; by Lemma 3.3, fupi g is bounded in H01 ðOÞ-C 1þa ðOÞ: It follows from the Ascoli–Arzela theorem that there exists a subsequence of pi (still % For any denoted by pi ) and u0 AH01 ðOÞ such that upi -u0 strongly in C 1 ðOÞ: 1 jAH0 ðOÞ; Z

rupi rj ¼ l

O

Z O

ug pi j þ

Z O

uppii j:

ð3:11Þ

Letting i-N and using Lebesgue’s theorem, we obtain Z O

ru0 rj ¼ l

Z O

ug 0 j

þ

Z u0 j: O

Thus u0 is the unique solution of (1.5) and jup  u0 jC 1 ðOÞ % -0 as p-1: We prove (iii) by contradiction. Suppose that jvp jLN ðOÞ pCo þ N as p-1: % and up ðxÞ; vp ðxÞXe distðx; @OÞ for xAO; Since up ; vp AC 2 ðOÞ-C 1þa ðOÞ u2p v2p 2 1 % vp AC ðOÞ-C ðOÞ:

0p ¼ ¼ ¼ p

u2p v2p up ;

Then

2  2 Z      rup  up rvp  þrvp  vp rup      vp up O Z  Dup Dvp  þ ðu2p  v2p Þ up vp O Z 2 2 ½lðug1  vg1 Þ þ ðup1  vp1 p p p p Þ ðup  vp Þ O Z ½lðg þ 1Þyg2 þ ðp  1Þyp2 ðup  vp Þ2 ðup þ vp Þ ðup pypvp Þ O Z ½lðg þ 1Þ þ ðp  1ÞC pþg ug2 ðup  vp Þ2 ðup þ vp Þ p O

o0

ð3:12Þ

as p-1:

A contradiction is obtained. So jvp jLN ðOÞ - þ N as p-1: Furthermore, by Lemma 3.1, jjvp jj- þ N as p-1: From the proof of Lemma 3.3, it is not hard to see that cn

there exists cn > 0 such that ðp  1Þlnjvp jLN ðOÞ Xcn ; i.e. jvp jLN ðOÞ Xep1 : The proof of this theorem is completed. &

Y. Haitao / J. Differential Equations 189 (2003) 487–512

511

Remark 3.3. From the proof of Theorem 2, ul;p is the minimal solution of (1.2) as p-1; and the result of Theorem 2 (iii) holds for any nonminimal solution of (1.2). To see this, we only need to show the existence of the minimal solution. In fact, let wl be the solution of (2.14), and consider the iteration: u0 ¼ wl and p Dun þ lug n þ un1 ¼ 0

in O;

un > 0 un ¼ 0

in O; on @O:

ð3:13Þ

Note that the functional 1 l F ðuÞ ¼ jjujj2  2 1g

Z O

juj

1g



Z O

upn1 u

is bounded below, coercive and lower semicontinuous on H01 ðOÞ: By the results in [14], inf uAH 1 ðOÞ F ðuÞ is attainable, and then (3.13) has a unique solution. Similar to 0 the proofs of Lemmas 2.3 and 3.1, it can be showed by induction that un1 pun pu in % So un O for any solution u of (1.2), and fun g is bounded in H01 ðOÞ-C 1þa ðOÞ: converges to the minimal solution of (1.2) as n-N:

Acknowledgment The author thanks Professor S.P. Wu for many helpful discussions.

References [1] R.P. Agarwal, D.O. Regan, Singular boundary value problems for superlinear second order ordinary and differential equations, J. Differential Equations 130 (1996) 333–355. [2] S. Alama, Semilinear elliptic equations with sublinear indefinite nonlinearities, Adv. in Differential Equations 4 (1999) 813–842. [3] A. Ambrosetti, J.G. Azorero, I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996) 219–242. [4] A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519–543. [5] M. Badiale, G. Tarantello, Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities, Nonlinear Anal. Theory Methods Appl. 29 (1997) 639–677. [6] H. Brezis, E. Lieb, A relations between pointwise convergence of functions and convergence of integrals, Proc. Amer. Math. Soc. 88 (1983) 486–490. [7] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equation involving the critical Sobolev exponent, Comm. Pure Appl. Math. 36 (1983) 437–477. [8] H. Brezis, L. Nirenberg, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris 317 (1993) 465–472. [9] H. Brezis, L. Oswald, Remarks on sublinear elliptic equation, Nonlinear Anal. Theory, Methods Appl. 10 (1986) 55–64. [10] M.M. Coclite, G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations 14 (1989) 1315–1327.

512

Y. Haitao / J. Differential Equations 189 (2003) 487–512

[11] M.G. Crandall, P.H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977) 193–222. [12] A. Edelson, Entire solutions of singular elliptic equations, J. Math. Anal. Appl. 139 (1989) 523–532. [13] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition, Springer, Berlin, 1983. [14] A.V. Lair, A.W. Shaker, Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl. 211 (1997) 371–385. [15] A.V. Lair, A.W. Shaker, Entire solution of a singular semilinear elliptic problem, J. Math. Anal. Appl. 200 (1996) 498–505. [16] A.C. Lazer, P.J. Mckenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc. 111 (1991) 721–730. [17] J.R. Lee, Asymptotic behavior of positive solutions of the equation Du ¼ lup as p-1; Comm. Partial Differential Equations 20 (1995) 633–646. [18] W.M. Ni, X.B. Pan, I. Takagi, Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents, Duke Math. J. 67 (1992) 1–20. [19] E.S. Noussair, J.F. Yang, On the existence of multiple positive solutions of critical semilinear elliptic problems, Nonlinear Anal. Theory, Methods Appl. 26 (1996) 1323–1346. [20] T. Ouyang, Solutions of semilinear elliptic equations Du þ lu þ hup ¼ 0 on compact manifolds, Part II, Indiana Univ. Math. J. 40 (1991) 1083–1140. [21] A.W. Shaker, On singular semilinear elliptic equations, J. Math. Anal. Appl. 173 (1993) 222–228. [22] J. Shi, M. Yao, On a singular nonlinear elliptic problem, Proc. Roy. Soc. Edinburg Sect. A 128 (1998) 1389–1401. [23] M. Struwe, Variational Methods, Springer, Berlin, 1990. [24] Y.J. Sun, S.P. Wu, Y.M. Long, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differential Equations 176 (2001) 511–531. [25] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincare´ Anal. Non Line´aire 9 (1992) 281–304. [26] Z.Q. Wang, The effect of the domain geometry on the number of positive solutions of Neumann problems with critical exponents, Differential Integral Equations 8 (1995) 1533–1554.