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Existence of positive boundary blow-up solutions for quasilinear elliptic equations via sub and supersolutions
Applied Mathematics and Computation 188 (2007) 492–498 www.elsevier.com/locate/amc
Existence of positive boundary blow-up solutions for quasilinear elliptic equations via sub and supersolutions Zuodong Yang a
a,b,*
, Bing Xu a, Mingzhu Wu
a
School of Mathematics and Computer Sciences, Nanjing Normal University, Jiangsu Nanjing 210097, China b School of Zhongbei, Nanjing Normal University, Jiangsu Nanjing 210046, China
Abstract In this paper, the existence of positive boundary blow-up weak solution for the quasilinear elliptic equation div(j$ujp2$u) = m(x)f(u) in a smooth bounded domain X RN or global space X = RN are obtained under new conditions. Our proof is based on the method of sub and supersolutions. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Quasilinear elliptic equation; Boundary blow-up; Sub and supersolutions; Keller–Osserman condition; Comparison principle
1. Introduction We consider the boundary blow-up elliptic problem ( divðjrujp2 ruÞ ¼ mðxÞf ðuÞ; x 2 X; ujoX ¼ 1;
ð1Þ
where p > 1, X RN, N P 2 or X = RN, and m is a Ho¨lder continuous positive function defined in X and f is locally Ho¨lder in (0, +1). By a positive boundary blow-up weak solutions u(x) of (1), we mean that 1 u 2 W 1;p loc ðXÞ \ C loc ðXÞ and u satisfies Z Z jrujp2 ru rw dx ¼ mðxÞf ðuÞw dx 8w 2 C 1 0 ðXÞ X
X
and u > 0 in X, u(x) ! 1 as d(x) ! 0, where d(x) = dist(x, oX). Such problems arise in the study of the subsonic motion of a gas [1], the electric potential in some bodies [2], and Riemannian geometry [3].
* Corresponding author. Present address: School of Mathematics and Computer Sciences, Nanjing Normal University, Jiangsu Nanjing 210097, China. E-mail address: [email protected] (Z. Yang).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.10.010
Z. Yang et al. / Applied Mathematics and Computation 188 (2007) 492–498
Boundary blow-up solutions of the problem DuðxÞ ¼ f ðuðxÞÞ; x 2 X; ujoX ¼ 1;
493
ð2Þ
where X is a bounded domain in RN(N P 1) have been extensively studied, see [4–13]. A problem with f(u) = eu and N = 2 was first considered by Bieberbach [6] in 1916. Bieberbach showed that if X is a bounded domain in R2 such that oX is a C2 submanifold of R2, then there exists a unique u 2 C2(X) such that Du = eu in X and ju(x) ln(d(x))2j is bounded on X. Here d(x) denotes the distance from a point x to oX. Rademacher [10], using the idea of Bieberbach, extended the above result to a smooth bounded domain in R3. In this case, the problem plays an important role, when N = 2, in the theory of Riemann surfaces of constant negative curvature and in the theory of automorphic functions, and when N = 3, according to [10], in the study of the electric potential in a glowing hollow metal body. Lazer and McKenna [2] extended the results for a bounded domain X in RN(N P 1) satisfying a uniform external sphere condition and the non-linearity f = f(x, u) = p(x)eu, where p(x) is continuous and strictly positive on X. Very recently, Lazer and McKenna [5] obtained similar results when D is replaced by the Monge–Ampere operator and X is a smooth, strictly convex, bounded domain. Similar results were also obtained for f = p(x)ua with a > 1. Posteraro [9], for f(u) = eu and N P 2, proved the estimates for the solution u(x) of Eq. (2) and for the measure of X comparing with a problem of the same type defined in a ball. In particular, when N = 2, Posteraro [9] obtained an explicit estimate of the minimum of u(x) in terms of the measure of X: min uðxÞ P lnð8p=jXjÞ: X
The existence, but not uniqueness, of solutions of Eq. (2) with f monotone was studied by Keller [11]. For f(u) = ua with a > 1, the problem (2) is of interest in the study of the sub-sonic motion of a gas when a = 2 (see [1]) and is related to a problem involving super-diffusion, particularly for 1 < a 6 2 (see [13,14]). Pohozaev [1] proved the existence, but not uniqueness, for the problem (2) when f(u) = u2. For the case where f(u) = u(N+2)/(N2)(N > 2), Loewener and Nirenberg [12] proved that if oX consists of a disjoint union of finitely compact C1 manifolds, each having co-dimension less than N/2 + 1, then there exists a unique solution of the problem (2). The uniqueness was established for f(u) = ua with a > 3, when oX is a C2-submanifold and D is replaced by a more general second-order elliptic operator, by Kondrat’ev and Nikishkin [11]. Marcus and Veron [7] proved the uniqueness for f(u) = ua with a > 1, when oX is compact and is locally the graph of a continuous function defined on an (N 1)-dimensional space. The existence of positive boundary blow-up solutions of (1) have been studied in previous papers (see [4,15,19]). In [4], Diaz and Letelier study the existence and uniqueness of boundary blow-up solutions to Eq. (1) both for f(u) = uc, c > p 1 (super-linear case) and X being a bounded domain were proved. Very recently, Lu, Yang and Twizell [15] and Yang [19] proved the existence of boundary blow-up solutions to Eq. (1) both for f(u) = uc, c > p 1, X = RN or X being a bounded domain (super-linear case) and c 6 p 1, X = RN (sub-linear case) respectively. In this paper, we obtain more results under new conditions. When p = 2, the related results have been obtained by Garcia-Melian [16]. The main results of the present paper complement the results in [4,15,16,19]. 2. Preliminaries We first consider quasilinear elliptic equation divðjruj
p2
ruÞ ¼ mðxÞf ðuÞ;
x 2 X RN ðN P 2Þ;
where p > 1, $u = ($1u, . . ., $Nu), m(x):X ! (0, 1) and f:(0, 1) ! (0, 1) are continuous functions. Throughout this section, we make the following assumptions without further mention. (H1) f:(0, 1) ! (0, 1) is locally Lipschitz continuous and non-decreasing (H2) f satisfy Keller–Osserman condition 1=p Z 1 Z u f ðsÞ ds du < 1: 1
0
ð3Þ
494
Z. Yang et al. / Applied Mathematics and Computation 188 (2007) 492–498
An important special case of (3) satisfying the above hypotheses is the equality p2
divðjruj
ruÞ ¼ mðxÞuc ;
x 2 X;
where c > p 1. From reference [17], we give the following lemma: Lemma 2.1 (Weak comparison principle). Let X be a bounded domain in RN(N P 2) with smooth boundary oX and h:(0, 1) ! (0, 1) is continuous and nondecreasing. Let u1,u2 2 W1,p(X) satisfy Z Z Z Z p2 p2 jru1 j ru1 rwdx þ hðu1 Þwdx 6 jru2 j ru2 rwdx þ hðu2 Þwdx; ð4Þ X
X
for all non-negative w 2 W u1 6 u 2
1;p 0 ðXÞ.
X
X
Then the inequality
on oX
implies that u1 6 u 2
in X:
From reference [15,18], we give the following lemma: Lemma 2.2. Let f(u) satisfy the following conditions: (i) f(s) is a single-value real continuous function defined for all real values of s and there exists a positive non-decreasing continuous function F(s) such that f(s) P F(s) and 1=p Z 1 Z x F ðzÞdz dx < 1; x0
0
for some x0 > 0 (ii) m(x) P b > 0 for x 2 RN, and u be a solution of divðjrujp2 ruÞ ¼ mðxÞf ðuÞ;
x 2 RN ðN P 2Þ;
in a domain D RN and continuous on its boundary S. Then there exists a decreasing function g(R) determined by F(u) such that uðP Þ 6 gðRðP ÞÞ:
ð5Þ
Here P denotes a point in D and R(P) denotes its distance from S. The function g(R) has the limits gðRÞ ! 1 as R ! 0; gðRÞ ! 1 as R ! 1:
ð6Þ ð7Þ
Lemma 2.3. If f(u) is non-decreasing and satisfies Lemma 2.2, then in any bounded domain D there exists a solution of (3) which becomes infinite on S. Proof. We note that for any constant a and any domain D there exists in D a solution ua of (3) which is equal to a on S, provided that f(u) is non-decreasing (see [17]). Furthermore, at each point of D, ua increases with a. If f(u) satisfies condition (i) of Lemma 2.2, then Lemma 2.2 holds, and at each point P in D all of the ua are bounded above. Thus in every closed subdomain, ua converges uniformly to a limit u. This limit is also a solution of (3). As P approaches S, u(P) increases infinitely, since on S, ua = a becomes infinite. Thus u is the desired solution and Lemma 2.3 is proved. h Lemma 2.4. Suppose f is non-decreasing and satisfies (H2). Then Z 1 ds < 1: 1=ðp1Þ f ðsÞ 1
ð8Þ
Z. Yang et al. / Applied Mathematics and Computation 188 (2007) 492–498
495
Proof. If we can prove that there exist positive numbers d and M such that f 1=ðp1Þ ðsÞ P dp s
ð9Þ
for s P M;
then we will be done since Z s f p=ðp1Þ ðsÞ F ðsÞ ¼ f ðtÞ dt 6 sf ðsÞ 6 dp 0
for s P M;
1=p
d P f 1=ðp1Þ for s P M so that (H2) implies (8). To prove (9), we assume it is which, in turn, yields ðF ðsÞÞ ðsÞ false. That is, we assume there exists an increasing sequence {sj} of real numbers such that limj!1sj = 1 and f1/(p1)(sj)/sj < 1/j for all j. Since f is increasing, we have f(s) 6 f(sj) for all s 2 [0, sj], which, in turn, produces F(s) 6 sf(s) 6 sf(sj) for s 2 [0, sj]. Hence,
ðp1Þ=p R R sj sj 1=p 1=p 1=p j ½F ðsÞ ds P ½sf ðs Þ ds P s ds j s1 s1 s1 sj ðp1Þ=p ðp1Þ=p ðp1Þ=p ðp1Þ=p ðp1Þ=p p j p ðsj s1 Þ ¼ p1 ðjÞ ð1 ðs1 =sj Þ Þ ! 1; ¼ p1 sj
R sj
as j ! 1, contradicting (H2). Thus (9) must be true. This completes the proof.
h
3. Main Results In this section, we state and prove our results. Throughout this section, we require the nonnegative function m and f to the following conditions: (A) Assume that m 2 Cm(X) for some 0 < m < 1, m > 0 in X, and there exist constants C1, C2 > 0 and c2 P c1 > p such that c
c
x 2 X:
C 2 dðxÞ 2 6 mðxÞ 6 C 1 dðxÞ 1 ;
ð10Þ
m
(B) Assume that f 2 C (0, +1) for some 0 < m < 1, and there exist q1 P q2 > p 1 such that f ðuÞ 6 C 1 uq1 u 2 R;
f ðuÞ P C 2 uq2
for large u:
ð11Þ
By a modification of the method given in [16], we obtain the following results. Theorem 3.1. Assume m and f verify hypotheses (10) and (11). Then problem (1) has at least a positive weak solution u which verifies D1 dðxÞ
a1
6 uðxÞ 6 D2 dðxÞ
a2
in X;
ð12Þ
where ai = (p + ci)/(qi p + 1), i = 1, 2, and D1, D2 are positive constants. First we state and prove an adaptation 1 of the method of sub and supersolutions to problem (1). A function u 2 W 1;p loc ðXÞ \ C loc ðXÞ is a subsolution to problem (1) if u = +1 on oX and Z Z mðxÞf ðuÞwdx 8w 2 C 1 jrujp2 ru rwdx P 0 ðXÞ: X
X
is a supersolution if Similarly, u u ¼ þ1 on oX and Z Z p2 jruj ru rwdx 6 mðxÞf ðuÞwdx 8w 2 C 1 0 ðXÞ: X
X
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Z. Yang et al. / Applied Mathematics and Computation 188 (2007) 492–498
Lemma 3.2. Assume there exist a subsolution u and a supersolution u to the problem (1) such that u 6 u. Then there exists at least a solution u to the problem (1) such that u 6 u 6 u. Proof. For n 2 N, we introduce the domain Xn = {x 2 X:d(x) > 1/n}, and consider the problem ( divðjrujp2 ruÞ ¼ mðxÞf ðuÞ; x 2 Xn ; ujoXn ¼ u:
ð13Þ
u a supersolution, the problem (13) has at least a positive weak solutions un such Since u is a subsolution and that u 6 un 6 u. This in particular gives local bounds for the sequence {un} which in turn leads to local bounds in C1,m. Thus for every k 2 N, we can select a subsequence fukn g which converges in C 1 ðXk Þ. A diagonal procedure gives a subsequence(denoted again by {un}) which converges to a function u in C 1loc ðXÞ. Passing to the limit in (13) we see that u is a weak solutions of the equation in (1), verifying u 6 u 6 u. In particular, we deduce that u = +1 on oX. This proves the lemma. h Proof of Theorem 3.1. First consider the problem ( divðjrU jp2 rU Þ ¼ U ri ; x 2 X; ujoX ¼ þ1
on oX;
where ri = p 1 + p/ai > p 1. From Lemma 2.3 and [4], this problem has a unique positive solution Ui such a a that CdðxÞ i 6 U i ðxÞ 6 C 0 dðxÞ i , for some positive constants C and C 0 . We set u = kU1, then u will be a subsolution provided that kp1 U r11 P mðxÞf ðkU 1 Þ: By hypothesis (11) on f, this is a consequence of k 6 ðC 1 supX mðxÞU 1 ðxÞq1 r1 Þ1=ðq1 pþ1Þ , which holds for small k, q r if the supermum is finite. But note that mðxÞU 11 1 6 CdðxÞr1 a1 ðq1 r1 Þ ¼ C in virtue of hypotheses (10), and the claim follows. In a similar way, we can see that u ¼ KU 2 is a supersolution for K P q r 1=ðq2 pþ1Þ ðC 2 inf X mðxÞU 22 2 Þ . Since a1 6 a2, it also follows that kU1 6 KU2, and Lemma 3.2 shows that there exist at least a positive weak solution to (1), which in addition verifies the estimates (12). This proves the theorem. For X = RN, we give the following theorem. h Theorem 3.3. Suppose f satisfies (H2), f 2 C1[0, 1], f(0) = 0, f 0 (u) P 0 and m(jxj) P C > 0 for x 2 RN and the following: 1=ðp1Þ Z 1 Z t 1N N 1 t s mðsÞ ds dt < 1: 0
0
Then p2
divðjruj
ruÞ ¼ mðjxjÞf ðuÞ
has an entire boundary blow-up positive radial solution. Proof. From Lemma 2.3, we have that for each k 2 N, the boundary-value problem p2
divðjrvk j rvk Þ ¼ mðjxjÞf ðvk Þ; vk ðxÞ ! 1 as jxj ! k
jxj < k;
has a positive solution. Furthermore, v1 P v2 P P vk P vkþ1 P > 0 N
in R . To prove our result, we need only prove.
h
Z. Yang et al. / Applied Mathematics and Computation 188 (2007) 492–498
497
(A) There exists w 2 C(RN), w > 0 such that vk P w in RN for all k and (B) v ! 1 as jxj ! 1, where v = limk!1vk. To prove (A), from theorem condition implies 1=ðp1Þ Z t Z r 1N N 1 zðrÞ ¼ C t s mðsÞ ds dt; 0
0
where C¼
Z
1
t1N
Z
0
t
sN 1 mðsÞ ds
1=ðp1Þ dt;
0
is the unique positive solution of the following problem divðjrzj
p2
rzÞ ¼ mðrÞ;
z ! 0 as r ¼ jxj ! 1;
x 2 RN :
By Lemma 2.4, we can define Z 1 dt for all s > 0: F ðsÞ ¼ 1=ðp1Þ f ðtÞ s Note also that F 0 ðsÞ ¼
1 f 1=ðp1Þ ðsÞ
<0
and
F 00 ðsÞ ¼
f 0 ðsÞ p > 0: ðp 1Þðf 1=ðp1Þ ðsÞÞ
A simple calculation shows that divðjrF ðvk Þj
p2
P jF 0 ðvk Þj
rF ðvk ÞÞ ¼ jF 0 ðvk Þj
p1
p1
divðjrvk j
p2
p2
rvk Þ þ ðp 1ÞjF 0 ðvk Þj
F 00 ðvk Þjrvk j
p
mðrÞf ðvk Þ ¼ mðrÞ:
Thus divðjrF ðvk Þj
p2
rF ðvk ÞÞ 6 divðjrzj
p2
rzÞ in jxj < k
and from Lemma 2.1, we obtain F(vk) 6 z if jxj 6 k. Let w = F1(z) and note that vk P w in RN. Consequently, v P w in RN and (A) is proved since w ! 1 as jxj ! 1 (here using the fact that lims!0F1(s) = 1). It is clear that (B) follows easily from (A). Acknowledgements Project Supported by the National Natural Science Foundation of China (No.10571022); the Natural Science Foundation of Educational Department of Jiangsu Province (No. 04KJB110062; No. 06KJB110056) and the Science Foundation of Nanjing Normal University (No. 2003SXXXGQ2B37). References [1] S.L. Pohozaev, The Dirichlet problem for the equation Du = u2, Soviet Math. Dokl. 1 (1960) 1143–1146. [2] A.C. Lazer, P.J. McKenna, On a problem of Bieberbach and Rademacher, Nonlinear Anal. 21 (1993) 327–335. [3] K. -S Cheng, W.-M. Ni, On the structure of the conformal scalar curvature equation on RN, Indiana Univ. Math. J. 41 (1992) 261– 278. [4] G. Diaz, R. Letelier, Explosive solutions of quasilinear elliptic equations: existence and uniqueness, Nonlinear Anal. 20 (1993) 97–125. [5] A.C. Lazer, P.J. McKenna, On singular boundary value problems for the Monge–Ampere operator, J. Math. Anal. Appl. 197 (1996) 341–362. [6] L. Bieberbach, Du = eu und die automorphen Funktionen, Math. Ann. 77 (1916) 173–212. [7] M. Marcus, L. Veron, Uniqueness of solutions with blow-up at the boundary for a class of nonlinear elliptic equation, C.R. Acad. Sci. Paris 317 (1993) 559–563. [8] M.R. Posteraro, On the solutions of the equation Du = eu blowing up on the boundary, C.R. Acad. Sci. Paris 322 (1996) 445–450.
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[9] H. Rademacher, Einige besondere problem partieller differentialgleichungen, in: Die Differential und Integralgleichungen, Second ed., der Mechanik und Physikl, Rosenberg, New York, 1943, pp. 838–845. [10] J.B. Keller, On solutions of Du = f(u), Commun. Pure Appl. Math. 10 (1957) 503–510. [11] V.A. Kondrat’ev, V.A. Nikishken, Asymptotics, near the boundary, of a singular boundary-value problem for a semilinear elliptic equation, Diff. Urav. 26 (1990) 465–468, English translation: Differential Equations 26(1990) 345–348. [12] C. Loewner, L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in: Contributions to Analysis (A Collection of Paper Dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 245–272. [13] E.B. Dynkin, Superprocesses and partial differential equations, Ann. Probab. 21 (1993) 1185–1262. [14] E.B. Dynkin, S.E. Kuznetsov, Superdiffusions and removable singularities for quasilinear partial differential equations, Commun. Pure Appl. Math. 49 (1996) 125–176. [15] Lu Qishao, Yang Zuodong, E.H. Twizell, Existence of entire explosive positive solutions of quasilinear elliptic equations, Appl. Math. Comput. 148 (2004) 359–372. [16] J. Garcia-Melian, A remark on the existence of large solutions via sub and supersolutions, E.J. Diff. Equat. 2003 (110) (2003) 1–4. [17] Z.M. Guo, Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problem, Nonlinear Anal. 18 (1992) 957–971. [18] Zuodong Yang, Existence of entire explosive positive radial solutions for a class of quasilinear elliptic systems, J. Math. Anal. Appl. 288 (2003) 768–783. [19] Zuodong Yang, Existence of explosive positive solutions of quasilinear elliptic equations, Appl. Math. Comput. 177 (2) (2006) 581–588.
Existence of positive boundary blow-up solutions for quasilinear elliptic equations via sub and supersolutions Zuodong Yang a
a,b,*
, Bing Xu a, Mingzhu Wu
a
School of Mathematics and Computer Sciences, Nanjing Normal University, Jiangsu Nanjing 210097, China b School of Zhongbei, Nanjing Normal University, Jiangsu Nanjing 210046, China
Abstract In this paper, the existence of positive boundary blow-up weak solution for the quasilinear elliptic equation div(j$ujp2$u) = m(x)f(u) in a smooth bounded domain X RN or global space X = RN are obtained under new conditions. Our proof is based on the method of sub and supersolutions. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Quasilinear elliptic equation; Boundary blow-up; Sub and supersolutions; Keller–Osserman condition; Comparison principle
1. Introduction We consider the boundary blow-up elliptic problem ( divðjrujp2 ruÞ ¼ mðxÞf ðuÞ; x 2 X; ujoX ¼ 1;
ð1Þ
where p > 1, X RN, N P 2 or X = RN, and m is a Ho¨lder continuous positive function defined in X and f is locally Ho¨lder in (0, +1). By a positive boundary blow-up weak solutions u(x) of (1), we mean that 1 u 2 W 1;p loc ðXÞ \ C loc ðXÞ and u satisfies Z Z jrujp2 ru rw dx ¼ mðxÞf ðuÞw dx 8w 2 C 1 0 ðXÞ X
X
and u > 0 in X, u(x) ! 1 as d(x) ! 0, where d(x) = dist(x, oX). Such problems arise in the study of the subsonic motion of a gas [1], the electric potential in some bodies [2], and Riemannian geometry [3].
* Corresponding author. Present address: School of Mathematics and Computer Sciences, Nanjing Normal University, Jiangsu Nanjing 210097, China. E-mail address: [email protected] (Z. Yang).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.10.010
Z. Yang et al. / Applied Mathematics and Computation 188 (2007) 492–498
Boundary blow-up solutions of the problem DuðxÞ ¼ f ðuðxÞÞ; x 2 X; ujoX ¼ 1;
493
ð2Þ
where X is a bounded domain in RN(N P 1) have been extensively studied, see [4–13]. A problem with f(u) = eu and N = 2 was first considered by Bieberbach [6] in 1916. Bieberbach showed that if X is a bounded domain in R2 such that oX is a C2 submanifold of R2, then there exists a unique u 2 C2(X) such that Du = eu in X and ju(x) ln(d(x))2j is bounded on X. Here d(x) denotes the distance from a point x to oX. Rademacher [10], using the idea of Bieberbach, extended the above result to a smooth bounded domain in R3. In this case, the problem plays an important role, when N = 2, in the theory of Riemann surfaces of constant negative curvature and in the theory of automorphic functions, and when N = 3, according to [10], in the study of the electric potential in a glowing hollow metal body. Lazer and McKenna [2] extended the results for a bounded domain X in RN(N P 1) satisfying a uniform external sphere condition and the non-linearity f = f(x, u) = p(x)eu, where p(x) is continuous and strictly positive on X. Very recently, Lazer and McKenna [5] obtained similar results when D is replaced by the Monge–Ampere operator and X is a smooth, strictly convex, bounded domain. Similar results were also obtained for f = p(x)ua with a > 1. Posteraro [9], for f(u) = eu and N P 2, proved the estimates for the solution u(x) of Eq. (2) and for the measure of X comparing with a problem of the same type defined in a ball. In particular, when N = 2, Posteraro [9] obtained an explicit estimate of the minimum of u(x) in terms of the measure of X: min uðxÞ P lnð8p=jXjÞ: X
The existence, but not uniqueness, of solutions of Eq. (2) with f monotone was studied by Keller [11]. For f(u) = ua with a > 1, the problem (2) is of interest in the study of the sub-sonic motion of a gas when a = 2 (see [1]) and is related to a problem involving super-diffusion, particularly for 1 < a 6 2 (see [13,14]). Pohozaev [1] proved the existence, but not uniqueness, for the problem (2) when f(u) = u2. For the case where f(u) = u(N+2)/(N2)(N > 2), Loewener and Nirenberg [12] proved that if oX consists of a disjoint union of finitely compact C1 manifolds, each having co-dimension less than N/2 + 1, then there exists a unique solution of the problem (2). The uniqueness was established for f(u) = ua with a > 3, when oX is a C2-submanifold and D is replaced by a more general second-order elliptic operator, by Kondrat’ev and Nikishkin [11]. Marcus and Veron [7] proved the uniqueness for f(u) = ua with a > 1, when oX is compact and is locally the graph of a continuous function defined on an (N 1)-dimensional space. The existence of positive boundary blow-up solutions of (1) have been studied in previous papers (see [4,15,19]). In [4], Diaz and Letelier study the existence and uniqueness of boundary blow-up solutions to Eq. (1) both for f(u) = uc, c > p 1 (super-linear case) and X being a bounded domain were proved. Very recently, Lu, Yang and Twizell [15] and Yang [19] proved the existence of boundary blow-up solutions to Eq. (1) both for f(u) = uc, c > p 1, X = RN or X being a bounded domain (super-linear case) and c 6 p 1, X = RN (sub-linear case) respectively. In this paper, we obtain more results under new conditions. When p = 2, the related results have been obtained by Garcia-Melian [16]. The main results of the present paper complement the results in [4,15,16,19]. 2. Preliminaries We first consider quasilinear elliptic equation divðjruj
p2
ruÞ ¼ mðxÞf ðuÞ;
x 2 X RN ðN P 2Þ;
where p > 1, $u = ($1u, . . ., $Nu), m(x):X ! (0, 1) and f:(0, 1) ! (0, 1) are continuous functions. Throughout this section, we make the following assumptions without further mention. (H1) f:(0, 1) ! (0, 1) is locally Lipschitz continuous and non-decreasing (H2) f satisfy Keller–Osserman condition 1=p Z 1 Z u f ðsÞ ds du < 1: 1
0
ð3Þ
494
Z. Yang et al. / Applied Mathematics and Computation 188 (2007) 492–498
An important special case of (3) satisfying the above hypotheses is the equality p2
divðjruj
ruÞ ¼ mðxÞuc ;
x 2 X;
where c > p 1. From reference [17], we give the following lemma: Lemma 2.1 (Weak comparison principle). Let X be a bounded domain in RN(N P 2) with smooth boundary oX and h:(0, 1) ! (0, 1) is continuous and nondecreasing. Let u1,u2 2 W1,p(X) satisfy Z Z Z Z p2 p2 jru1 j ru1 rwdx þ hðu1 Þwdx 6 jru2 j ru2 rwdx þ hðu2 Þwdx; ð4Þ X
X
for all non-negative w 2 W u1 6 u 2
1;p 0 ðXÞ.
X
X
Then the inequality
on oX
implies that u1 6 u 2
in X:
From reference [15,18], we give the following lemma: Lemma 2.2. Let f(u) satisfy the following conditions: (i) f(s) is a single-value real continuous function defined for all real values of s and there exists a positive non-decreasing continuous function F(s) such that f(s) P F(s) and 1=p Z 1 Z x F ðzÞdz dx < 1; x0
0
for some x0 > 0 (ii) m(x) P b > 0 for x 2 RN, and u be a solution of divðjrujp2 ruÞ ¼ mðxÞf ðuÞ;
x 2 RN ðN P 2Þ;
in a domain D RN and continuous on its boundary S. Then there exists a decreasing function g(R) determined by F(u) such that uðP Þ 6 gðRðP ÞÞ:
ð5Þ
Here P denotes a point in D and R(P) denotes its distance from S. The function g(R) has the limits gðRÞ ! 1 as R ! 0; gðRÞ ! 1 as R ! 1:
ð6Þ ð7Þ
Lemma 2.3. If f(u) is non-decreasing and satisfies Lemma 2.2, then in any bounded domain D there exists a solution of (3) which becomes infinite on S. Proof. We note that for any constant a and any domain D there exists in D a solution ua of (3) which is equal to a on S, provided that f(u) is non-decreasing (see [17]). Furthermore, at each point of D, ua increases with a. If f(u) satisfies condition (i) of Lemma 2.2, then Lemma 2.2 holds, and at each point P in D all of the ua are bounded above. Thus in every closed subdomain, ua converges uniformly to a limit u. This limit is also a solution of (3). As P approaches S, u(P) increases infinitely, since on S, ua = a becomes infinite. Thus u is the desired solution and Lemma 2.3 is proved. h Lemma 2.4. Suppose f is non-decreasing and satisfies (H2). Then Z 1 ds < 1: 1=ðp1Þ f ðsÞ 1
ð8Þ
Z. Yang et al. / Applied Mathematics and Computation 188 (2007) 492–498
495
Proof. If we can prove that there exist positive numbers d and M such that f 1=ðp1Þ ðsÞ P dp s
ð9Þ
for s P M;
then we will be done since Z s f p=ðp1Þ ðsÞ F ðsÞ ¼ f ðtÞ dt 6 sf ðsÞ 6 dp 0
for s P M;
1=p
d P f 1=ðp1Þ for s P M so that (H2) implies (8). To prove (9), we assume it is which, in turn, yields ðF ðsÞÞ ðsÞ false. That is, we assume there exists an increasing sequence {sj} of real numbers such that limj!1sj = 1 and f1/(p1)(sj)/sj < 1/j for all j. Since f is increasing, we have f(s) 6 f(sj) for all s 2 [0, sj], which, in turn, produces F(s) 6 sf(s) 6 sf(sj) for s 2 [0, sj]. Hence,
ðp1Þ=p R R sj sj 1=p 1=p 1=p j ½F ðsÞ ds P ½sf ðs Þ ds P s ds j s1 s1 s1 sj ðp1Þ=p ðp1Þ=p ðp1Þ=p ðp1Þ=p ðp1Þ=p p j p ðsj s1 Þ ¼ p1 ðjÞ ð1 ðs1 =sj Þ Þ ! 1; ¼ p1 sj
R sj
as j ! 1, contradicting (H2). Thus (9) must be true. This completes the proof.
h
3. Main Results In this section, we state and prove our results. Throughout this section, we require the nonnegative function m and f to the following conditions: (A) Assume that m 2 Cm(X) for some 0 < m < 1, m > 0 in X, and there exist constants C1, C2 > 0 and c2 P c1 > p such that c
c
x 2 X:
C 2 dðxÞ 2 6 mðxÞ 6 C 1 dðxÞ 1 ;
ð10Þ
m
(B) Assume that f 2 C (0, +1) for some 0 < m < 1, and there exist q1 P q2 > p 1 such that f ðuÞ 6 C 1 uq1 u 2 R;
f ðuÞ P C 2 uq2
for large u:
ð11Þ
By a modification of the method given in [16], we obtain the following results. Theorem 3.1. Assume m and f verify hypotheses (10) and (11). Then problem (1) has at least a positive weak solution u which verifies D1 dðxÞ
a1
6 uðxÞ 6 D2 dðxÞ
a2
in X;
ð12Þ
where ai = (p + ci)/(qi p + 1), i = 1, 2, and D1, D2 are positive constants. First we state and prove an adaptation 1 of the method of sub and supersolutions to problem (1). A function u 2 W 1;p loc ðXÞ \ C loc ðXÞ is a subsolution to problem (1) if u = +1 on oX and Z Z mðxÞf ðuÞwdx 8w 2 C 1 jrujp2 ru rwdx P 0 ðXÞ: X
X
is a supersolution if Similarly, u u ¼ þ1 on oX and Z Z p2 jruj ru rwdx 6 mðxÞf ðuÞwdx 8w 2 C 1 0 ðXÞ: X
X
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Lemma 3.2. Assume there exist a subsolution u and a supersolution u to the problem (1) such that u 6 u. Then there exists at least a solution u to the problem (1) such that u 6 u 6 u. Proof. For n 2 N, we introduce the domain Xn = {x 2 X:d(x) > 1/n}, and consider the problem ( divðjrujp2 ruÞ ¼ mðxÞf ðuÞ; x 2 Xn ; ujoXn ¼ u:
ð13Þ
u a supersolution, the problem (13) has at least a positive weak solutions un such Since u is a subsolution and that u 6 un 6 u. This in particular gives local bounds for the sequence {un} which in turn leads to local bounds in C1,m. Thus for every k 2 N, we can select a subsequence fukn g which converges in C 1 ðXk Þ. A diagonal procedure gives a subsequence(denoted again by {un}) which converges to a function u in C 1loc ðXÞ. Passing to the limit in (13) we see that u is a weak solutions of the equation in (1), verifying u 6 u 6 u. In particular, we deduce that u = +1 on oX. This proves the lemma. h Proof of Theorem 3.1. First consider the problem ( divðjrU jp2 rU Þ ¼ U ri ; x 2 X; ujoX ¼ þ1
on oX;
where ri = p 1 + p/ai > p 1. From Lemma 2.3 and [4], this problem has a unique positive solution Ui such a a that CdðxÞ i 6 U i ðxÞ 6 C 0 dðxÞ i , for some positive constants C and C 0 . We set u = kU1, then u will be a subsolution provided that kp1 U r11 P mðxÞf ðkU 1 Þ: By hypothesis (11) on f, this is a consequence of k 6 ðC 1 supX mðxÞU 1 ðxÞq1 r1 Þ1=ðq1 pþ1Þ , which holds for small k, q r if the supermum is finite. But note that mðxÞU 11 1 6 CdðxÞr1 a1 ðq1 r1 Þ ¼ C in virtue of hypotheses (10), and the claim follows. In a similar way, we can see that u ¼ KU 2 is a supersolution for K P q r 1=ðq2 pþ1Þ ðC 2 inf X mðxÞU 22 2 Þ . Since a1 6 a2, it also follows that kU1 6 KU2, and Lemma 3.2 shows that there exist at least a positive weak solution to (1), which in addition verifies the estimates (12). This proves the theorem. For X = RN, we give the following theorem. h Theorem 3.3. Suppose f satisfies (H2), f 2 C1[0, 1], f(0) = 0, f 0 (u) P 0 and m(jxj) P C > 0 for x 2 RN and the following: 1=ðp1Þ Z 1 Z t 1N N 1 t s mðsÞ ds dt < 1: 0
0
Then p2
divðjruj
ruÞ ¼ mðjxjÞf ðuÞ
has an entire boundary blow-up positive radial solution. Proof. From Lemma 2.3, we have that for each k 2 N, the boundary-value problem p2
divðjrvk j rvk Þ ¼ mðjxjÞf ðvk Þ; vk ðxÞ ! 1 as jxj ! k
jxj < k;
has a positive solution. Furthermore, v1 P v2 P P vk P vkþ1 P > 0 N
in R . To prove our result, we need only prove.
h
Z. Yang et al. / Applied Mathematics and Computation 188 (2007) 492–498
497
(A) There exists w 2 C(RN), w > 0 such that vk P w in RN for all k and (B) v ! 1 as jxj ! 1, where v = limk!1vk. To prove (A), from theorem condition implies 1=ðp1Þ Z t Z r 1N N 1 zðrÞ ¼ C t s mðsÞ ds dt; 0
0
where C¼
Z
1
t1N
Z
0
t
sN 1 mðsÞ ds
1=ðp1Þ dt;
0
is the unique positive solution of the following problem divðjrzj
p2
rzÞ ¼ mðrÞ;
z ! 0 as r ¼ jxj ! 1;
x 2 RN :
By Lemma 2.4, we can define Z 1 dt for all s > 0: F ðsÞ ¼ 1=ðp1Þ f ðtÞ s Note also that F 0 ðsÞ ¼
1 f 1=ðp1Þ ðsÞ
<0
and
F 00 ðsÞ ¼
f 0 ðsÞ p > 0: ðp 1Þðf 1=ðp1Þ ðsÞÞ
A simple calculation shows that divðjrF ðvk Þj
p2
P jF 0 ðvk Þj
rF ðvk ÞÞ ¼ jF 0 ðvk Þj
p1
p1
divðjrvk j
p2
p2
rvk Þ þ ðp 1ÞjF 0 ðvk Þj
F 00 ðvk Þjrvk j
p
mðrÞf ðvk Þ ¼ mðrÞ:
Thus divðjrF ðvk Þj
p2
rF ðvk ÞÞ 6 divðjrzj
p2
rzÞ in jxj < k
and from Lemma 2.1, we obtain F(vk) 6 z if jxj 6 k. Let w = F1(z) and note that vk P w in RN. Consequently, v P w in RN and (A) is proved since w ! 1 as jxj ! 1 (here using the fact that lims!0F1(s) = 1). It is clear that (B) follows easily from (A). Acknowledgements Project Supported by the National Natural Science Foundation of China (No.10571022); the Natural Science Foundation of Educational Department of Jiangsu Province (No. 04KJB110062; No. 06KJB110056) and the Science Foundation of Nanjing Normal University (No. 2003SXXXGQ2B37). References [1] S.L. Pohozaev, The Dirichlet problem for the equation Du = u2, Soviet Math. Dokl. 1 (1960) 1143–1146. [2] A.C. Lazer, P.J. McKenna, On a problem of Bieberbach and Rademacher, Nonlinear Anal. 21 (1993) 327–335. [3] K. -S Cheng, W.-M. Ni, On the structure of the conformal scalar curvature equation on RN, Indiana Univ. Math. J. 41 (1992) 261– 278. [4] G. Diaz, R. Letelier, Explosive solutions of quasilinear elliptic equations: existence and uniqueness, Nonlinear Anal. 20 (1993) 97–125. [5] A.C. Lazer, P.J. McKenna, On singular boundary value problems for the Monge–Ampere operator, J. Math. Anal. Appl. 197 (1996) 341–362. [6] L. Bieberbach, Du = eu und die automorphen Funktionen, Math. Ann. 77 (1916) 173–212. [7] M. Marcus, L. Veron, Uniqueness of solutions with blow-up at the boundary for a class of nonlinear elliptic equation, C.R. Acad. Sci. Paris 317 (1993) 559–563. [8] M.R. Posteraro, On the solutions of the equation Du = eu blowing up on the boundary, C.R. Acad. Sci. Paris 322 (1996) 445–450.
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