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The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems
J. Math. Anal. Appl. 308 (2005) 532–540 www.elsevier.com/locate/jmaa
The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems ✩ Zhijun Zhang Department of Mathematics and Information Science, Yantai University, Yantai 264005, People’s Republic of China Received 15 April 2004 Available online 26 February 2005 Submitted by P.J. McKenna
Abstract By constructing the comparison functions and the perturbed method, it is showed that any solution u ∈ C 2 (Ω) to the semilinear elliptic problems ∆u = k(x)g(u), x ∈ Ω, u|∂Ω = +∞ satisfies (2+σ )(2+ρ+σ ) 1/ρ limd(x)→0 Z(du(x) , where Ω is a bounded domain with smooth boundary µ (x)) = 2c (2+ρ) 0
2+σ ; g ∈ C 1 [0, ∞), g 0 and g(s) is inin RN ; limd(x)→0 dk(x) σ (x) = c0 , −2 < σ , c0 > 0, µ = s 2 ∞ g (sξ ) dt ρ √ = s, creasing on (0, ∞), there exists ρ > 0 such that lims→∞ g (s) = ξ , ∀ξ > 0, Z(s) 2G(t) t G(t) = 0 g(s) ds. 2005 Elsevier Inc. All rights reserved.
Keywords: Semilinear elliptic equations; Large solutions; Precise asymptotic behaviour; Uniqueness
✩ This work is supported in part by National Natural Science Foundation of People’s Republic of China under Grant numbers 10071066, 10251002. E-mail addresses: [email protected], [email protected].
0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.11.029
Z. Zhang / J. Math. Anal. Appl. 308 (2005) 532–540
533
1. Introduction The purpose of this paper is to investigate the precise asymptotic behaviour of the solutions near the boundary to the model problems ∆u = k(x)g(u),
x ∈ Ω,
u|∂Ω = +∞,
(1.1)
where the last condition means that u(x) → +∞ as d(x) = dist(x, ∂Ω) → 0, and the solution is called ‘large solution’ or ‘explosive solution,’ Ω is a bounded domain with smooth boundary in RN (N 1), g ∈ C 1 [0, ∞) and satisfies (g1 ) g(0) = 0, g 0 on [0, ∞); (g2 ) Keller–Osserman condition: ∞ s
dt < ∞, √ 2G(t)
t ∀s > 0, G(t) =
g(s) ds; 0
α (Ω) for some α ∈ (0, 1), and is nonnegative in Ω and positive near ∂Ω. Moreover, k ∈ Cloc it may be singular or zero on ∂Ω. The problems go back to the pioneering Bieberbach’s work in 1916 and Rademacher’s work in 1943 (see, for example, [13]) where k(x) ≡ 1, g(u) = eu , N = 2 and N = 3. The problems arise from many branches of mathematics and applied mathematics; and have been discussed by many authors and in many contexts; see, for instance, [1–7,9–17,19, 21–23]. For k(x) ≡ 1, Osserman–Keller [10,19] supplied a necessary and sufficient condition (g2 ) for the existence of large solutions. Then, by analyzing the corresponding ordinary differential equation, combining with the maximum principle, Bandle and Marcus [1] established the following results: if g satisfies (g1 ) and
(g3 ) there exist θ > 0 and t0 1 such that g(ξ t) ξ 1+θ g(t) for all ξ ∈ (0, 1) and t t0 /ξ , then for any solution u of problem (1.1) u(x) → 1 as d(x) → 0, Z(d(x)) where Z satisfies Z (s) = g Z(s) ,
s ∈ (0, b),
Z(s) → +∞,
as s → 0.
Moreover, in addition to the conditions given above, g satisfies (g4 ) g(sξ ) ξg(s) for all ξ ∈ (0, 1) and all s 0, then problem (1.1) has a unique solution. Moreover, Lazer–McKenna [14] showed that if g satisfies (g1 ) and
(1.2)
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Z. Zhang / J. Math. Anal. Appl. 308 (2005) 532–540
(g5 ) there exists a1 > 0 such that g (s) is nondecreasing for s a1 , and g (s) lim √ = ∞, s→∞ G(s) then for any solution u of problem (1.1) u(x) − Z(d(x)) → 0 as d(x) → 0. Most recently, applying the regularly varying functions and their properties, which were first introduced and established by Karamata in 1930, and constructing the comparison functions, Cirstea and Radulescu [3–5] showed that if g satisfies (g1 ) and the following conditions: (g6 ) g(s)/s is increasing on (0, ∞); (g7 ) there exists ρ > 0 such that lims→∞
g (sξ ) g (s)
= ξ ρ , ∀ξ > 0,
¯ k 0 in Ω, k = 0 on ∂Ω, and satisfies the following assumptions: there and k ∈ C α (Ω), exist δ0 > 0 and a positive increasing function h ∈ C 1 (0, δ0 ) such that k(x) = c0 > 0; 2 t h (d(x)) 0t h(s)ds h(s)ds d =l limt→0+ 0 h(t) = 0 and limt→0+ dt h(t)
(k1 ) limd(x)→0+ (k2 )
∈ [0, 1],
then any solution u of problem (1.1) satisfies u(x) = ξ0 , d(x)→0 Z(d(x)) 1/ρ and Z ∈ C 2 (0, a) (a ∈ (0, δ0 )) is defined by where ξ0 = c02+lρ (2+ρ) lim
∞ Z(t)
ds = √ 2G(s)
t h(s) ds,
∀t ∈ (0, a).
0
In this paper, applying the Cirstea and Radulescu’s argument, constructing the comparison functions, we show the following results in the case of limd(x)→0 dk(x) σ (x) = c0 with σ > −2 and c0 > 0. α (Ω) is nonnegative in Ω and Theorem 1.1. If g satisfies (g1 ), (g6 ), (g7 ), and k ∈ Cloc positive near ∂Ω, and limd(x)→0 dk(x) σ (x) = c0 , σ > −2, c0 > 0, then every solution u ∈ C 2 (Ω) to problem (1.1) satisfies (2 + σ )(2 + ρ + σ ) 1/ρ u(x) lim = , (1.3) d(x)→0 Z(d µ (x)) 2c0 (2 + ρ) ∞ √ dt where µ = 2+σ 2 and Z(s) 2G(t) = s, ∀s > 0.
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In particular, when g(u) = up , p > 1, ρ = p − 1, 2(p + 1) 1/(p−1) −2/(p−1) Z(s) = cs , c= , (p − 1)2 every solution u to problem (1.1) satisfies (2+σ )/(p−1) (2 + σ )(p + σ + 1) 1/(p−1) = . lim u(x) d(x) d(x)→0 c0 (p − 1)2 Remark 1.1. When g(u) = up , p > 1 and σ 0, by a different argument, López-Gómez [16] showed the same results as in Theorem 1.1 for the more general boundary condition.
2. Proof of Theorem 1.1 First we give some preliminary considerations. Let us recall some basic definitions and the properties to Karamata regular variation theory (see [18,20]). Definition 2.1. A positive measure function f defined on [a, ∞), for some a > 0, is called regularly varying at infinity with index ρ, written f ∈ Rρ , if for each ξ > 0 and some ρ ∈ R, f (ξ t) = ξρ. t→∞ f (t)
(2.1)
lim
The real number ρ is called the index of regular variation. Definition 2.2. A positive measure function L defined on [a, ∞), for some a > 0, is called slowly varying at infinity, if for each ξ > 0 L(ξ t) = 1. t→∞ L(t) It follows by the definitions that if f ∈ Rρ it can be represented in the form lim
(2.2)
f (u) = uρ L(u).
(2.3)
Lemma 2.1 (Uniform convergence theorem). If f ∈ Rρ , then (2.1) (and so (2.2)) holds uniformly for ξ ∈ [a, b] with 0 < a < b. Lemma 2.2 (Representation theorem). The function L is slowly varying at infinity if and only if it may be written in the form u
y(s) L(u) = c(u)exp ds , u a, (2.4) s a
for some a > 0, where c(u) and y(u) are measurable and for u → ∞, y(u) → 0 and c(u) → c, with c > 0.
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Z. Zhang / J. Math. Anal. Appl. 308 (2005) 532–540
Lemma 2.3 (Cirstea and Radulescu [3, Lemma 1]). Assume g satisfies (g1 ) and (g6 ), then the following are equivalent: sg (s) (ii) lim g ∈ Rρ ; = ρ + 1; s→∞ g(s) G(s) = (2 + ρ)−1 . (iii) lim s→∞ g(s) Moreover, if ρ > 0, then
u y(s) ρ+1 ds , lim g (s) = ∞ and g(u) = u exp s→∞ s (i)
(2.5)
(2.6)
0
where y ∈ C[0, ∞) is nonnegative such that lims→0+
y(s) s
∈ [0, ∞) and lims→∞ y(s) = 0.
Lemma 2.4. If g satisfies (g1 ), (g6 ) and (g7 ) with ρ > 0, then Z(t) in Theorem 1.1 has the following properties: lim
t→0+
Z (t) = 0 and Z (t)
lim
t→0+
ρ Z (t) =− . tZ (t) 2+ρ
Proof. It follows from the definition of Z(t) that
Z (t) = − 2G(Z(t)), Z (t) = g Z(t) ,
∀t > 0,
(2.7)
lim Z(t) = +∞,
t→0+
(2.8) and by Lemma 2.3 and l’Hospital’s rule, we see that 2 Z (t) 2G(u) 1 Z (t) = lim = lim = 0, i.e., lim = 0, lim u→∞ g 2 (u) u→∞ g (u) t→0+ Z (t) t→0+ Z (t) and
−1 ∞
Z (t) dv = − lim 2G(Z(t)) g Z(t) lim √ t→0+ tg(Z(t)) t→0+ 2G(v) √
∞
Z(t)
dv = − lim √ u→∞ 2G(v) u 2g (u)G(u) − 1 = − lim u→∞ g 2 (u) G(u) = 1 + 2 lim −1 u→∞ g(u) 1 −1 =1+2 2+ρ ρ =− . 2 2+ρ 2G(u) g(u)
−1
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Lemma 2.5 (Tao and Zhang [21, Theorem 1]). If g satisfies (g1 ), (g2 ), and k satisfies the condition in Theorem 1.1, then problem (1.1) has a solution u ∈ C 2 (Ω). Moreover, for any solution u ∈ C 2 (Ω), u(x) H −1 v(x) ¯ , ∀x ∈ Ω, ¯ is the unique solution to the problem where v¯ ∈ C 2+α (Ω) ∩ C(Ω) −∆v = k(x), v > 0, x ∈ Ω, v|∂Ω = 0, ∞ ds and H (t) = t f (s) , t > 0, H −1 denotes the inverse function of H .
(2.9)
Remark 2.1. If g satisfies (g1 ), (g7 ), then we see by (2.3) and (2.4) that g satisfies (g2 ) and the function H (t) is well-defined in (0, ∞). With the same proof as in [21, Theorem 1], we can show the following result. ¯ Ω¯ 0 to the problem Lemma 2.6. Lemma 2.5 still holds on Ω/ ∆u = k(x)g(u), u = +∞,
x ∈ Ω/Ω¯ 0 ,
u = 1,
x ∈ ∂Ω,
x ∈ ∂Ω0 ,
(2.10)
where Ω0 Ω and ∂Ω0 is a smooth submanifold of dimension N − 1, v¯ ∈ C 2+α (Ω/Ω¯ 0 ) ∩ ¯ Ω¯ 0 ) is the unique solution to the following problem: C(Ω/ −∆v(x) = k(x),
v > 0, x ∈ Ω/Ω¯ 0 ,
∞ v=
ds , f (s)
x ∈ ∂Ω,
1
v = 0,
x ∈ ∂Ω0 .
(2.11)
Proof of Theorem 1.1. For any δ > 0, we define Ωδ = {x ∈ Ω: d µ (x) < δ}, and ∂Ωδ = {x ∈ Ω: d µ (x) = δ}. By the regularity of ∂Ω, we can choose δ sufficiently small such that (i) d(x) ∈ C 2 (Ω¯ 2δ ); (ii) Z (s) µ(µ − 1) Z (s) + ρ + µd(x)∆ d(x) < ε, sZ (s) 2 + ρ sZ (s) for all (x, s) ∈ Ω2δ × (0, 2δ). Define
(2 + σ )(2 + ρ + σ ) 2c0 (2 + ρ) 4ε 1/ρ ρ , ξ1 = ξ0 − c0 c ξρ where ε ∈ 0, 04 0 . ξ0 =
1/ρ ,
4ε 1/ρ ρ ξ2 = ξ0 + , c0
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Let β ∈ (0, δ) be arbitrary. We define u¯ β = ξ2 Z(d µ (x) − β) for any x with β < < 2δ, and uβ = ξ1 Z(d µ (x) + β), for any x with d µ (x) + β < 2δ. It follows by |∇d(x)| = 1 that −∆u¯ β (x) + k(x)g u¯ β (x) = k(x)g ξ2 Z d µ (x) − β − µ2 ξ2 d 2µ−2 (x)Z d µ (x) − β − µ(µ − 1)ξ2 d µ−2 (x)Z d µ (x) − β − µξ2 d µ−1 (x)Z d µ (x) − β ∆d(x) k(x)g(ξ2 Z(d µ (x) − β)) = ξ2 d 2µ−2 (x)g Z d µ (x) − β ξ2 d σ (x)g(Z(d µ (x) − β)) Z (d µ (x) − β) ρµ(µ − 1) ρ − µ(µ − 1) µ − µ2 − + ρ+2 d (x)Z (d µ (x) − β) ρ + 2 Z (d µ (x) − β) d(x)µ∆d(x) − µ d (x)Z (d µ (x) − β) ρ ξ2 d σ (x)g Z d µ (x) − β c 0 ξ2 − ε Z (d µ (x) − β) ρ ρ − c0 ξ0 − µ(µ − 1) µ + d (x)Z (d µ (x) − β) ρ + 2 Z (d µ (x) − β) d(x)µ∆d(x) − µ d (x)Z (d µ (x) − β) 0 d µ (x)
and −∆uβ (x) + k(x)g uβ (x) = k(x)g ξ1 Z d µ (x) + β − µ2 ξ1 d 2µ−2 (x)Z d µ (x) + β − µ(µ − 1)ξ1 d µ−2 (x)Z d µ (x) + β − µξ1 d µ−1 (x)Z d µ (x) + β ∆d(x) k(x)g(ξ1 Z(d µ (x) + β)) = ξ1 d 2µ−2 (x)g Z d µ (x) + β ξ1 d σ (x)g(Z(d µ (x) + β)) Z (d µ (x) + β) ρ ρµ(µ − 1) − µ(µ − 1) µ + − µ2 − ρ+2 d (x)Z (d µ (x) + β) ρ + 2 Z (d µ (x) + β) d(x)µ∆d(x) − µ d (x)Z (d µ (x) + β) µ ρ σ ξ1 d (x)g Z d (x) + β c 0 ξ1 + ε ρ Z (d µ (x) + β) ρ − c0 ξ0 − µ(µ − 1) µ + d (x)Z (d µ (x) + β) ρ + 2 Z (d µ (x) + β) d(x)µ∆d(x) − µ d (x)Z (d µ (x) + β) 0.
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Now let w be an arbitrary solution of problem (2.10) with Ω0 = Ωδ and u be an arbitrary solution of problem (1.1) and v = u + w. We see that u + w|∂Ω = +∞ > uβ |∂Ω , u¯ β + w|∂Ωβ = +∞ > u|∂Ωβ ,
u + w|∂Ωδ = +∞ > uβ |∂Ωδ , u¯ β + w|∂Ωδ = +∞ > u|∂Ωδ .
It follows by (g1 ) and the comparison principle [8, Theorem 10.1] that uβ (x) u(x) + w(x),
∀x ∈ Ωδ ,
u(x) u¯ β (x) + w(x),
Let β → 0, we see that ξ1 Z d µ (x) u(x) + w(x) ξ2 Z d µ (x) + 2w(x),
∀x ∈ Ωδ ∩ Ωβ .
∀x ∈ Ωδ ,
which implies ξ1 lim inf d(x)→0
u(x) u(x) lim sup ξ2 . Z(d µ (x)) d(x)→0 Z(d µ (x))
Let ε → 0, and look at the definitions of ξ1 and ξ2 ; we have u(x) = ξ0 . d(x)→0 Z(d µ (x)) lim
Finally, as the same proof as in [1,3–5,14], we can show the uniqueness. Let u1 , u2 ∈ C 2 (Ω) be two solutions of problem (1.1). By (1.3), we see that limd(x)→0 u1 (x)/u2 (x) = 1. Hence, for any ε ∈ (0, 1), there exists δ > 0, such that (1 − ε)u2 (x) u1 (x) (1 + ε)u2 (x),
∀x ∈ Ωδ .
Moreover, we see by (g6 ) that for every x ∈ Ω, ∆(1 − ε)u2 (x) k(x)g (1 − ε)u2 (x) , ∆(1 + ε)u2 (x) k(x)g (1 + ε)u2 (x) . Thus the comparison principle [8, Theorem 10.1] implies that (1 − ε)u2 (x) u1 (x) (1 + ε)u2 (x), ∀x ∈ x ∈ Ω: d(x) δ/2 . Let ε → 0, we see that u1 ≡ u2 in Ω. The proof is finished.
2
References [1] C. Bandle, M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J. Anal. Math. 58 (1992) 9–24. [2] C. Bandle, E. Giarrusso, Boundary blowup for semilinear elliptic equations with nonlinear gradient terms, Adv. Differential Equations 1 (1996) 133–150. [3] F. Cirstea, V.D. Radulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorption, C. R. Acad. Sci. Paris Sér. I Math. 335 (2002) 447–452. [4] F. Cirstea, V.D. Radulescu, Asymptotics for the blow-up boundary solution of the logistic equation with absorption, C. R. Acad. Sci. Paris Sér. I Math. 336 (2003) 231–236. [5] F. Cirstea, V.D. Radulescu, Solutions with boundary blow-up for a class of nonlinear elliptic problems, Houston J. Math. 29 (2003) 821–829.
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[6] G. Diaz, R. Letelier, Explosive solutions of quasilinear elliptic equation: existence and uniqueness, Nonlinear Anal. 20 (1993) 97–125. [7] E. Giarrusso, Asymptotic behavior of large solutions of an elliptic quasilinear equation with a borderline case, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 777–782. [8] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, third ed., SpringerVerlag, Berlin, 1998. [9] A. Greco, G. Porru, Asymptotic estimates and convexity of large solutions to semilinear elliptic equations, Differential Integral Equations 10 (1997) 219–229. [10] J.B. Keller, On solution of ∆u = f (u), Comm. Pure Appl. Math. 10 (1957) 503–510. [11] A.V. Lair, A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations, J. Math. Anal. Appl. 240 (1999) 205–218. [12] J.M. Lasry, P.L. Lions, Nonlinear elliptic equation with singular boundary conditions and stochastic control with state constraints, Math. Z. 283 (1989) 583–630. [13] A.C. Lazer, P.J. McKenna, On a problem of Bieberbach and Rademacher, Nonlinear Anal. 21 (1993) 327– 335. [14] A.C. Lazer, P.J. McKenna, Asymptotic behavior of solutions of boundary blowup problems, Differential Integral Equations 7 (1994) 1001–1019. [15] C. Loewner, L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in: Contributions to Analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 245–272. [16] J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45. [17] M. Marcus, L. Véron, Uniqueness of solutions with blowup on the boundary for a class of nonlinear elliptic equations, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993) 557–563. [18] V. Maric, Regular Variation and Differential Equations, Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. [19] R. Osserman, On the inequality ∆u f (u), Pacific J. Math. 7 (1957) 1641–1647. [20] R. Seneta, Regular Varying Functions, Lecture Notes in Math., vol. 508, Springer-Verlag, Berlin, 1976. [21] S. Tao, Z. Zhang, On the existence of explosive solutions for semilinear elliptic problems, Nonlinear Anal. 48 (2002) 1043–1050. [22] L. Véron, Semilinear elliptic equations with uniform blowup on the boundary, J. Anal. Math. 59 (1992) 231–250. [23] Z. Zhang, A remark on the existence of explosive solutions for a class of semilinear elliptic equations, Nonlinear Anal. 41 (2000) 143–148.
The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems ✩ Zhijun Zhang Department of Mathematics and Information Science, Yantai University, Yantai 264005, People’s Republic of China Received 15 April 2004 Available online 26 February 2005 Submitted by P.J. McKenna
Abstract By constructing the comparison functions and the perturbed method, it is showed that any solution u ∈ C 2 (Ω) to the semilinear elliptic problems ∆u = k(x)g(u), x ∈ Ω, u|∂Ω = +∞ satisfies (2+σ )(2+ρ+σ ) 1/ρ limd(x)→0 Z(du(x) , where Ω is a bounded domain with smooth boundary µ (x)) = 2c (2+ρ) 0
2+σ ; g ∈ C 1 [0, ∞), g 0 and g(s) is inin RN ; limd(x)→0 dk(x) σ (x) = c0 , −2 < σ , c0 > 0, µ = s 2 ∞ g (sξ ) dt ρ √ = s, creasing on (0, ∞), there exists ρ > 0 such that lims→∞ g (s) = ξ , ∀ξ > 0, Z(s) 2G(t) t G(t) = 0 g(s) ds. 2005 Elsevier Inc. All rights reserved.
Keywords: Semilinear elliptic equations; Large solutions; Precise asymptotic behaviour; Uniqueness
✩ This work is supported in part by National Natural Science Foundation of People’s Republic of China under Grant numbers 10071066, 10251002. E-mail addresses: [email protected], [email protected].
0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.11.029
Z. Zhang / J. Math. Anal. Appl. 308 (2005) 532–540
533
1. Introduction The purpose of this paper is to investigate the precise asymptotic behaviour of the solutions near the boundary to the model problems ∆u = k(x)g(u),
x ∈ Ω,
u|∂Ω = +∞,
(1.1)
where the last condition means that u(x) → +∞ as d(x) = dist(x, ∂Ω) → 0, and the solution is called ‘large solution’ or ‘explosive solution,’ Ω is a bounded domain with smooth boundary in RN (N 1), g ∈ C 1 [0, ∞) and satisfies (g1 ) g(0) = 0, g 0 on [0, ∞); (g2 ) Keller–Osserman condition: ∞ s
dt < ∞, √ 2G(t)
t ∀s > 0, G(t) =
g(s) ds; 0
α (Ω) for some α ∈ (0, 1), and is nonnegative in Ω and positive near ∂Ω. Moreover, k ∈ Cloc it may be singular or zero on ∂Ω. The problems go back to the pioneering Bieberbach’s work in 1916 and Rademacher’s work in 1943 (see, for example, [13]) where k(x) ≡ 1, g(u) = eu , N = 2 and N = 3. The problems arise from many branches of mathematics and applied mathematics; and have been discussed by many authors and in many contexts; see, for instance, [1–7,9–17,19, 21–23]. For k(x) ≡ 1, Osserman–Keller [10,19] supplied a necessary and sufficient condition (g2 ) for the existence of large solutions. Then, by analyzing the corresponding ordinary differential equation, combining with the maximum principle, Bandle and Marcus [1] established the following results: if g satisfies (g1 ) and
(g3 ) there exist θ > 0 and t0 1 such that g(ξ t) ξ 1+θ g(t) for all ξ ∈ (0, 1) and t t0 /ξ , then for any solution u of problem (1.1) u(x) → 1 as d(x) → 0, Z(d(x)) where Z satisfies Z (s) = g Z(s) ,
s ∈ (0, b),
Z(s) → +∞,
as s → 0.
Moreover, in addition to the conditions given above, g satisfies (g4 ) g(sξ ) ξg(s) for all ξ ∈ (0, 1) and all s 0, then problem (1.1) has a unique solution. Moreover, Lazer–McKenna [14] showed that if g satisfies (g1 ) and
(1.2)
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Z. Zhang / J. Math. Anal. Appl. 308 (2005) 532–540
(g5 ) there exists a1 > 0 such that g (s) is nondecreasing for s a1 , and g (s) lim √ = ∞, s→∞ G(s) then for any solution u of problem (1.1) u(x) − Z(d(x)) → 0 as d(x) → 0. Most recently, applying the regularly varying functions and their properties, which were first introduced and established by Karamata in 1930, and constructing the comparison functions, Cirstea and Radulescu [3–5] showed that if g satisfies (g1 ) and the following conditions: (g6 ) g(s)/s is increasing on (0, ∞); (g7 ) there exists ρ > 0 such that lims→∞
g (sξ ) g (s)
= ξ ρ , ∀ξ > 0,
¯ k 0 in Ω, k = 0 on ∂Ω, and satisfies the following assumptions: there and k ∈ C α (Ω), exist δ0 > 0 and a positive increasing function h ∈ C 1 (0, δ0 ) such that k(x) = c0 > 0; 2 t h (d(x)) 0t h(s)ds h(s)ds d =l limt→0+ 0 h(t) = 0 and limt→0+ dt h(t)
(k1 ) limd(x)→0+ (k2 )
∈ [0, 1],
then any solution u of problem (1.1) satisfies u(x) = ξ0 , d(x)→0 Z(d(x)) 1/ρ and Z ∈ C 2 (0, a) (a ∈ (0, δ0 )) is defined by where ξ0 = c02+lρ (2+ρ) lim
∞ Z(t)
ds = √ 2G(s)
t h(s) ds,
∀t ∈ (0, a).
0
In this paper, applying the Cirstea and Radulescu’s argument, constructing the comparison functions, we show the following results in the case of limd(x)→0 dk(x) σ (x) = c0 with σ > −2 and c0 > 0. α (Ω) is nonnegative in Ω and Theorem 1.1. If g satisfies (g1 ), (g6 ), (g7 ), and k ∈ Cloc positive near ∂Ω, and limd(x)→0 dk(x) σ (x) = c0 , σ > −2, c0 > 0, then every solution u ∈ C 2 (Ω) to problem (1.1) satisfies (2 + σ )(2 + ρ + σ ) 1/ρ u(x) lim = , (1.3) d(x)→0 Z(d µ (x)) 2c0 (2 + ρ) ∞ √ dt where µ = 2+σ 2 and Z(s) 2G(t) = s, ∀s > 0.
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In particular, when g(u) = up , p > 1, ρ = p − 1, 2(p + 1) 1/(p−1) −2/(p−1) Z(s) = cs , c= , (p − 1)2 every solution u to problem (1.1) satisfies (2+σ )/(p−1) (2 + σ )(p + σ + 1) 1/(p−1) = . lim u(x) d(x) d(x)→0 c0 (p − 1)2 Remark 1.1. When g(u) = up , p > 1 and σ 0, by a different argument, López-Gómez [16] showed the same results as in Theorem 1.1 for the more general boundary condition.
2. Proof of Theorem 1.1 First we give some preliminary considerations. Let us recall some basic definitions and the properties to Karamata regular variation theory (see [18,20]). Definition 2.1. A positive measure function f defined on [a, ∞), for some a > 0, is called regularly varying at infinity with index ρ, written f ∈ Rρ , if for each ξ > 0 and some ρ ∈ R, f (ξ t) = ξρ. t→∞ f (t)
(2.1)
lim
The real number ρ is called the index of regular variation. Definition 2.2. A positive measure function L defined on [a, ∞), for some a > 0, is called slowly varying at infinity, if for each ξ > 0 L(ξ t) = 1. t→∞ L(t) It follows by the definitions that if f ∈ Rρ it can be represented in the form lim
(2.2)
f (u) = uρ L(u).
(2.3)
Lemma 2.1 (Uniform convergence theorem). If f ∈ Rρ , then (2.1) (and so (2.2)) holds uniformly for ξ ∈ [a, b] with 0 < a < b. Lemma 2.2 (Representation theorem). The function L is slowly varying at infinity if and only if it may be written in the form u
y(s) L(u) = c(u)exp ds , u a, (2.4) s a
for some a > 0, where c(u) and y(u) are measurable and for u → ∞, y(u) → 0 and c(u) → c, with c > 0.
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Lemma 2.3 (Cirstea and Radulescu [3, Lemma 1]). Assume g satisfies (g1 ) and (g6 ), then the following are equivalent: sg (s) (ii) lim g ∈ Rρ ; = ρ + 1; s→∞ g(s) G(s) = (2 + ρ)−1 . (iii) lim s→∞ g(s) Moreover, if ρ > 0, then
u y(s) ρ+1 ds , lim g (s) = ∞ and g(u) = u exp s→∞ s (i)
(2.5)
(2.6)
0
where y ∈ C[0, ∞) is nonnegative such that lims→0+
y(s) s
∈ [0, ∞) and lims→∞ y(s) = 0.
Lemma 2.4. If g satisfies (g1 ), (g6 ) and (g7 ) with ρ > 0, then Z(t) in Theorem 1.1 has the following properties: lim
t→0+
Z (t) = 0 and Z (t)
lim
t→0+
ρ Z (t) =− . tZ (t) 2+ρ
Proof. It follows from the definition of Z(t) that
Z (t) = − 2G(Z(t)), Z (t) = g Z(t) ,
∀t > 0,
(2.7)
lim Z(t) = +∞,
t→0+
(2.8) and by Lemma 2.3 and l’Hospital’s rule, we see that 2 Z (t) 2G(u) 1 Z (t) = lim = lim = 0, i.e., lim = 0, lim u→∞ g 2 (u) u→∞ g (u) t→0+ Z (t) t→0+ Z (t) and
−1 ∞
Z (t) dv = − lim 2G(Z(t)) g Z(t) lim √ t→0+ tg(Z(t)) t→0+ 2G(v) √
∞
Z(t)
dv = − lim √ u→∞ 2G(v) u 2g (u)G(u) − 1 = − lim u→∞ g 2 (u) G(u) = 1 + 2 lim −1 u→∞ g(u) 1 −1 =1+2 2+ρ ρ =− . 2 2+ρ 2G(u) g(u)
−1
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Lemma 2.5 (Tao and Zhang [21, Theorem 1]). If g satisfies (g1 ), (g2 ), and k satisfies the condition in Theorem 1.1, then problem (1.1) has a solution u ∈ C 2 (Ω). Moreover, for any solution u ∈ C 2 (Ω), u(x) H −1 v(x) ¯ , ∀x ∈ Ω, ¯ is the unique solution to the problem where v¯ ∈ C 2+α (Ω) ∩ C(Ω) −∆v = k(x), v > 0, x ∈ Ω, v|∂Ω = 0, ∞ ds and H (t) = t f (s) , t > 0, H −1 denotes the inverse function of H .
(2.9)
Remark 2.1. If g satisfies (g1 ), (g7 ), then we see by (2.3) and (2.4) that g satisfies (g2 ) and the function H (t) is well-defined in (0, ∞). With the same proof as in [21, Theorem 1], we can show the following result. ¯ Ω¯ 0 to the problem Lemma 2.6. Lemma 2.5 still holds on Ω/ ∆u = k(x)g(u), u = +∞,
x ∈ Ω/Ω¯ 0 ,
u = 1,
x ∈ ∂Ω,
x ∈ ∂Ω0 ,
(2.10)
where Ω0 Ω and ∂Ω0 is a smooth submanifold of dimension N − 1, v¯ ∈ C 2+α (Ω/Ω¯ 0 ) ∩ ¯ Ω¯ 0 ) is the unique solution to the following problem: C(Ω/ −∆v(x) = k(x),
v > 0, x ∈ Ω/Ω¯ 0 ,
∞ v=
ds , f (s)
x ∈ ∂Ω,
1
v = 0,
x ∈ ∂Ω0 .
(2.11)
Proof of Theorem 1.1. For any δ > 0, we define Ωδ = {x ∈ Ω: d µ (x) < δ}, and ∂Ωδ = {x ∈ Ω: d µ (x) = δ}. By the regularity of ∂Ω, we can choose δ sufficiently small such that (i) d(x) ∈ C 2 (Ω¯ 2δ ); (ii) Z (s) µ(µ − 1) Z (s) + ρ + µd(x)∆ d(x) < ε, sZ (s) 2 + ρ sZ (s) for all (x, s) ∈ Ω2δ × (0, 2δ). Define
(2 + σ )(2 + ρ + σ ) 2c0 (2 + ρ) 4ε 1/ρ ρ , ξ1 = ξ0 − c0 c ξρ where ε ∈ 0, 04 0 . ξ0 =
1/ρ ,
4ε 1/ρ ρ ξ2 = ξ0 + , c0
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Let β ∈ (0, δ) be arbitrary. We define u¯ β = ξ2 Z(d µ (x) − β) for any x with β < < 2δ, and uβ = ξ1 Z(d µ (x) + β), for any x with d µ (x) + β < 2δ. It follows by |∇d(x)| = 1 that −∆u¯ β (x) + k(x)g u¯ β (x) = k(x)g ξ2 Z d µ (x) − β − µ2 ξ2 d 2µ−2 (x)Z d µ (x) − β − µ(µ − 1)ξ2 d µ−2 (x)Z d µ (x) − β − µξ2 d µ−1 (x)Z d µ (x) − β ∆d(x) k(x)g(ξ2 Z(d µ (x) − β)) = ξ2 d 2µ−2 (x)g Z d µ (x) − β ξ2 d σ (x)g(Z(d µ (x) − β)) Z (d µ (x) − β) ρµ(µ − 1) ρ − µ(µ − 1) µ − µ2 − + ρ+2 d (x)Z (d µ (x) − β) ρ + 2 Z (d µ (x) − β) d(x)µ∆d(x) − µ d (x)Z (d µ (x) − β) ρ ξ2 d σ (x)g Z d µ (x) − β c 0 ξ2 − ε Z (d µ (x) − β) ρ ρ − c0 ξ0 − µ(µ − 1) µ + d (x)Z (d µ (x) − β) ρ + 2 Z (d µ (x) − β) d(x)µ∆d(x) − µ d (x)Z (d µ (x) − β) 0 d µ (x)
and −∆uβ (x) + k(x)g uβ (x) = k(x)g ξ1 Z d µ (x) + β − µ2 ξ1 d 2µ−2 (x)Z d µ (x) + β − µ(µ − 1)ξ1 d µ−2 (x)Z d µ (x) + β − µξ1 d µ−1 (x)Z d µ (x) + β ∆d(x) k(x)g(ξ1 Z(d µ (x) + β)) = ξ1 d 2µ−2 (x)g Z d µ (x) + β ξ1 d σ (x)g(Z(d µ (x) + β)) Z (d µ (x) + β) ρ ρµ(µ − 1) − µ(µ − 1) µ + − µ2 − ρ+2 d (x)Z (d µ (x) + β) ρ + 2 Z (d µ (x) + β) d(x)µ∆d(x) − µ d (x)Z (d µ (x) + β) µ ρ σ ξ1 d (x)g Z d (x) + β c 0 ξ1 + ε ρ Z (d µ (x) + β) ρ − c0 ξ0 − µ(µ − 1) µ + d (x)Z (d µ (x) + β) ρ + 2 Z (d µ (x) + β) d(x)µ∆d(x) − µ d (x)Z (d µ (x) + β) 0.
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Now let w be an arbitrary solution of problem (2.10) with Ω0 = Ωδ and u be an arbitrary solution of problem (1.1) and v = u + w. We see that u + w|∂Ω = +∞ > uβ |∂Ω , u¯ β + w|∂Ωβ = +∞ > u|∂Ωβ ,
u + w|∂Ωδ = +∞ > uβ |∂Ωδ , u¯ β + w|∂Ωδ = +∞ > u|∂Ωδ .
It follows by (g1 ) and the comparison principle [8, Theorem 10.1] that uβ (x) u(x) + w(x),
∀x ∈ Ωδ ,
u(x) u¯ β (x) + w(x),
Let β → 0, we see that ξ1 Z d µ (x) u(x) + w(x) ξ2 Z d µ (x) + 2w(x),
∀x ∈ Ωδ ∩ Ωβ .
∀x ∈ Ωδ ,
which implies ξ1 lim inf d(x)→0
u(x) u(x) lim sup ξ2 . Z(d µ (x)) d(x)→0 Z(d µ (x))
Let ε → 0, and look at the definitions of ξ1 and ξ2 ; we have u(x) = ξ0 . d(x)→0 Z(d µ (x)) lim
Finally, as the same proof as in [1,3–5,14], we can show the uniqueness. Let u1 , u2 ∈ C 2 (Ω) be two solutions of problem (1.1). By (1.3), we see that limd(x)→0 u1 (x)/u2 (x) = 1. Hence, for any ε ∈ (0, 1), there exists δ > 0, such that (1 − ε)u2 (x) u1 (x) (1 + ε)u2 (x),
∀x ∈ Ωδ .
Moreover, we see by (g6 ) that for every x ∈ Ω, ∆(1 − ε)u2 (x) k(x)g (1 − ε)u2 (x) , ∆(1 + ε)u2 (x) k(x)g (1 + ε)u2 (x) . Thus the comparison principle [8, Theorem 10.1] implies that (1 − ε)u2 (x) u1 (x) (1 + ε)u2 (x), ∀x ∈ x ∈ Ω: d(x) δ/2 . Let ε → 0, we see that u1 ≡ u2 in Ω. The proof is finished.
2
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