Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 19, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
BOUNDARY BEHAVIOR OF SOLUTIONS TO A SINGULAR
DIRICHLET PROBLEM WITH A NONLINEAR CONVECTION
BO LI, ZHIJUN ZHANG
Abstract. In this article we analyze the exact boundary behavior of solutions
to the singular nonlinear Dirichlet problem
−∆u = b(x)g(u) + λ|∇u|q + σ,
˛
u˛
= 0,
u > 0, x ∈ Ω,
∂Ω
where Ω is a bounded domain with smooth boundary in RN , q ∈ (0, 2], σ > 0,
λ > 0, g ∈ C 1 ((0, ∞), (0, ∞)), lims→0+ g(s) = ∞, g is decreasing on (0, s0 )
α (Ω) for some α ∈ (0, 1), is positive in Ω, but may be
for some s0 > 0, b ∈ Cloc
vanishing or singular on the boundary. We show that λ|∇u|q does not affect
the first expansion of classical solutions near the boundary.
1. Introduction
In this article, we consider the boundary behavior of solutions to the singular
nonlinear Dirichlet problem
(1.1)
− ∆u = b(x)g(u) + λ|∇u|q + σ, u > 0, x ∈ Ω, u∂Ω = 0,
where Ω is a bounded domain with smooth boundary in RN , q ∈ (0, 2], λ > 0,
σ > 0, b satisfies
α
(B1) b ∈ Cloc
(Ω) for some α ∈ (0, 1), is positive in Ω,
and g satisfies
(G1) g ∈ C 1 ((0, ∞), (0, ∞)) and lims→0 g(s) = ∞;
(G2) there exists s0 > 0 such that g 0 (s) < 0, for all s ∈ (0, s0 );
(G3) there exists Cg ≥ 0 such that
Z s
dτ
0
lim g (s)
= −Cg .
g(τ
)
s→0+
0
A typical example of functions which satisfy (G1)-(G3) is
g(s) = s−γ + µsp , s > 0,
where γ, p, µ > 0. In this case, Cg = γ/(1 + γ). A complete characterization of g
in (G1)-(G3) is provided in Lemma 2.14.
2000 Mathematics Subject Classification. 35J65, 35B05, 35J25, 60J50.
Key words and phrases. Semilinear elliptic equation; singular Dirichlet problem;
nonlinear convection term; classical solution; boundary behavior.
c
2015
Texas State University - San Marcos.
Submitted December 3, 2014. Published January 20, 2015.
1
2
B. LI, Z. ZHANG
For convenience, we denote by ψ the solution to the problem
Z ψ(t)
ds
= t, ∀t > 0.
g(s)
0
EJDE-2015/19
(1.2)
When λ = 0, (1.1) arises in the study of non-Newtonian fluids, boundary layer
phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the
theory of heat conduction in electrical materials (see, for instance, [9, 17, 21, 36, 38,
41]) and has been discussed by many authors and in many contexts. With regard
to the existence, nonexistence, uniqueness, multiplicity, regularity, local (near the
boundary) and global estimates of (classical or weak) solutions, see, for instance, [1][4], [6, 8, 9, 12, 16, 17, 21], [23]-[27], [29]-[31], [35], [39]-[46], [51] and the references
therein.
When λ > 0, b ≡ 1 in Ω and g(u) = u−γ with γ > 0, the authors [47] considered
the existence and regularities of the unique solution to (1.1). Cui [11] established
a sub-supersolution method to more general problem than (1.1).
When λ = 1, σ = 0, 0 < q < 2, b ≡ 1 in Ω and the function g : (0, ∞) → (0, ∞) is
locally Lipschitz continuous and decreasing, Giarrusso and Porru [19] showed that
if g satisfies the following conditions
R1
R∞
(G01) 0 g(s)ds = ∞, 1 g(s)ds < ∞;
(G02) there exist positive constants δ and
R ∞M with M > 1 such that G(s) <
M G(2s), for all s ∈ (0, δ), G(s) := s g(τ )dτ , s > 0,
then the unique solution u to (1.1) has the following properties:
(I1) |u(x) − φ(d(x))| < c0 d(x), for all x ∈ Ω for 0 < q ≤ 1;
(I2) |u(x) − φ(d(x))| < c0 d(x)[G(φ(d(x)))](q−1)/2 , for all x ∈ Ω for 1 < q < 2;
where d(x) = dist(x, ∂Ω), c0 is a suitable positive constant and φ ∈ C[0, ∞) ∩
C 2 (0, ∞) is the unique solution of the problem
Z φ(t)
ds
p
= t, t > 0.
(1.3)
2G(s)
0
For further works, see [10], [13]-[15], [18, 20, 28, 37], [48]-[50] and the references
therein.
We introduce two types of functions. First, we denote by K the set of all Karamata functions L̂ which are normalized slowly varying at zero (see, BinghamGoldie-Teugels’s book [5] and Maric’s book [32]) defined on (0, η] for some η > 0
by
Z η y(τ ) L̂(s) = c0 exp
dτ , s ∈ (0, η],
(1.4)
τ
s
where c0 > 0 and the function y ∈ C([0, η]) with y(0) = 0.
Next let Λ denote the set of all positive monotonic functions θ in C 1 (0, δ0 ) ∩
1
L (0, δ0 ) (δ0 > 0) which satisfy
Z t
d Θ(t) lim
:= Cθ ∈ [0, ∞), Θ(t) :=
θ(s)ds.
(1.5)
t→0 dt θ(t)
0
The set Λ was first introduced by Cı̂rstea and Rǎdulescu [7] for non-decreasing functions and by Mohammed [34] for non-increasing functions to study the boundary
behavior of solutions to boundary blow-up elliptic problems.
We assume that b satisfies
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BOUNDARY BEHAVIOR OF SOLUTIONS
3
(B2) there exists θ ∈ Λ such that
0 < b1 := lim inf
d(x)→0
b(x)
b(x)
≤ b2 := lim sup 2
< ∞.
θ2 (d(x))
θ (d(x))
d(x)→0
Recently, for g satisfying (G1) and decreasing on (0, ∞), the authors [50] considered
the two cases
(i) q ∈ (0, 2), b ≡ 1 in Ω, g satisfies (G3) with Cg > 1/2;
(ii) q = 2, b satisfies (B1) and (B2), g satisfies (G3) with
Cθ + 2Cg > 2,
(1.6)
and one of the following two conditions holds
(S01) Cg > 0;
(S02) Cg = 0 and λ lim sups→0+ |gg(s)
0 (s)| < 1
and obtained the boundary behavior of the unique solutions to (1.1).
In this article, we extend [50] for more general g and b. We first establish a local
comparison principle for q ∈ (0, 1) under (G2). More precisely, we show the first
exact asymptotic behaviour of any classical solution near the boundary to (1.1)
and reveal that the nonlinear gradient term λ|∇u|q does not affect the behaviour.
For q ∈ [1, 2], by using a nonlinear change, the local comparison principle and the
results in [51] and [30], we show the same results as q ∈ (0, 1). Our main results
are summarized as follows.
Theorem 1.1. For fixed λ > 0, let g satisfy (G1)–(G3), b satisfy (B1)–(B2). If
both (1.6) and one of the following conditions hold
(S1) q ∈ (0, 1);
(S2) q ∈ [1, 2] and Cg > 0;
(S3) q ∈ [1, 2], Cg = 0 and
λ lim sup
s→0+
g(s)
< 1,
|g 0 (s)|
then for any classical solution uλ to (1.1), it holds
1−Cg
ξ1
≤ lim inf
d(x)→0
where
ξ1 =
uλ (x)
uλ (x)
1−C
≤ lim sup
≤ ξ2 g ,
ψ(Θ2 (d(x))) d(x)→0
ψ(Θ2 (d(x)))
b1
,
2(Cθ + 2Cg − 2)
ξ2 =
b2
.
2(Cθ + 2Cg − 2)
In particular,
(i) when Cg = 1, uλ satisfies
lim
d(x)→0
uλ (x)
= 1;
ψ(Θ2 (d(x)))
(ii) when Cg < 1 and b1 = b2 = b0 in (B2), uλ satisfies
uλ (x)
2
d(x)→0 ψ(d (x)θ 2 (d(x)))
lim
where
ξ01 =
= (ξ01 Cθ2 )1−Cg ,
b0
.
2(Cθ + 2Cg − 2)
(1.7)
(1.8)
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B. LI, Z. ZHANG
EJDE-2015/19
Theorem 1.2. For fixed λ > 0, let q ∈ (0, 2], g satisfy (G1)–(G3), and let b satisfy
(B1) and
Rη
(B3) there exists L̂ ∈ K with 0 L̂(s)
s ds < ∞ such that
0 < b1 := lim inf
d(x)→0
b(x)
a2 (d(x))
≤ b2 := lim sup
d(x)→0
b(x)
a2 (d(x))
< ∞,
where
a2 (t) = t−2 L̂(t), t ∈ (0, η].
(1.9)
If one of (S1), (S2), (S3) holds, then for any classical solution uλ to (1.1), it holds
1−Cg
b1
≤ lim inf
d(x)→0
uλ (x)
uλ (x)
1−C
≤ lim sup
≤ b2 g ,
ψ(h1 (d(x))) d(x)→0
ψ(h1 (d(x)))
where
t
Z
h1 (t) =
0
L̂(s)
ds,
s
t ∈ (0, η).
(1.10)
(1.11)
In particular,
(i) when Cg = 1, uλ satisfies
uλ (x)
= 1;
d(x)→0 ψ(h1 (d(x)))
lim
(ii) when Cg < 1 and b1 = b2 = b0 in (B3 ), uλ satisfies
uλ (x)
1−C
= b0 g .
d(x)→0 ψ(h1 (d(x)))
lim
Theorem 1.3. For fixed λ > 0, let q ∈ (0, 2], b satisfy (B1), g satisfy (G1) and
g(s) = s−γ + µsp , s ∈ (0, s0 ), for some s0 > 0, where γ, p, µ > 0. If b satisfies
Rη
(B4) there exists L̂ ∈ K with 0 L̂(s)
s ds = ∞ such that
0 < b1 := lim inf
d(x)→0
b(x)
(d(x))γ−1 L̂(d(x))
≤ b2 := lim sup
d(x)→0
b(x)
(d(x))γ−1 L̂(d(x))
< ∞,
then for any classical solution uλ to (1.1), it holds
uλ (x)
d(x)(h2 (d(x)))1/(1+γ)
uλ (x)
≤ lim sup
d(x)→0
d(x)(h2 (d(x)))1/(1+γ)
(b1 (1 + γ))1/(1+γ) ≤ lim inf
d(x)→0
(1.12)
≤ (b2 (1 + γ))1/(1+γ) ,
where
Z
h2 (t) =
t
η
L(τ )
dτ,
τ
t ∈ (0, η).
(1.13)
Remark 1.4. Some basic examples of functions which satisfy (G1)–(G3) with
Cg = 0 and lims→0+ |gg(s)
0 (s)| = 0 are
(i) g(s) = (− ln s)γ , γ > 0, s ∈ (0, s0 );
(ii) g(s) = (ln(− ln s))γ , γ > 0, s ∈ (0, s0 );
γ
(iii) g(s) = e(− ln s) , 0 < γ < 1, s ∈ (0, s0 ), where s0 > 0 sufficiently small.
Remark 1.5. When γ > 0, we note that Cg =
i.e., Cθ + 2Cg = 2.
γ
1+γ
and Cθ =
2
γ+1
in Theorem 1.3,
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BOUNDARY BEHAVIOR OF SOLUTIONS
5
The outline of this paper is as follows. In section 2, we present some basic facts
from Karamata regular variation theory and some preliminaries. Some comparison
principles are given in section 3. In section 4, we prove Theorems 1.1–1.3.
2. Preliminaries
Our approach relies on Karamata regular variation theory established by Karamata in 1930 which is a basic tool in stochastic processes (see Bingham, Goldie and
Teugels’ book [5], Maric’s book [32] and the references therein). In this section, we
present some basic facts from Karamata regular variation theory.
Definition 2.1. A positive continuous function g defined on (0, η], for some η > 0,
is called regularly varying at zero with index ρ, denoted by g ∈ RV Zρ , if for
each ξ > 0 and some ρ ∈ R,
g(ξs)
= ξρ.
(2.1)
lim
s→0+ g(s)
In particular, when ρ = 0, g is called slowly varying at zero.
Clearly, if g ∈ RV Zρ , then L(s) := g(s)/sρ is slowly varying at zero.
Definition 2.2. A positive continuous function g defined on (0, η], for some η > 0,
is called rapidly varying to infinity at zero if for each ξ ∈ (0, 1)
lim
s→0+
g(ξs)
= ∞.
g(s)
(2.2)
Definition 2.3. A positive function g ∈ C(0, η] with lims→0+ g(s) = 0, for some
η > 0, is called rapidly varying to zero at zero if for each ξ ∈ (0, 1)
lim
s→0+
g(ξs)
= 0.
g(s)
(2.3)
Proposition 2.4 (Uniform convergence theorem). If g ∈ RV Zρ , then (2.1) holds
uniformly for ξ ∈ [c1 , c2 ] with 0 < c1 < c2 .
Proposition 2.5 (Representation theorem). A function L is slowly varying at zero
if and only if it may be written in the form
Z η y(τ ) dτ , s ∈ (0, η],
L(s) = l(s) exp
(2.4)
τ
s
where the functions l and y are continuous and for s → 0+ , y(s) → 0 and l(s) → c0 ,
with c0 > 0.
Note that
Z
η
y(τ ) dτ ,
τ
s
is normalized slowly varying at zero, and
L̂(s) = c0 exp
g(s) = sρ L̂(s),
s ∈ (0, η],
s ∈ (0, η],
(2.5)
(2.6)
is normalized regularly varying at zero with index ρ (and denoted by g ∈ N RV Zρ ).
A function g ∈ N RV Zρ if and only if
g ∈ C 1 (0, η], for some η > 0 and lim+
s→0
sg 0 (s)
= ρ.
g(s)
Proposition 2.6. If functions L, L1 are slowly varying at zero, then
(2.7)
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B. LI, Z. ZHANG
EJDE-2015/19
(i) Lρ for every ρ ∈ R, c1 L + c2 L1 (c1 ≥ 0, c2 ≥ 0 with c1 + c2 > 0), L · L1 ,
L ◦ L1 (if L1 (s) → 0 as s → 0+ ) , are also slowly varying at zero;
(ii) For every ρ > 0 and s → 0+ , sρ L(s) → 0, s−ρ L(s) → ∞;
(iii) For ρ ∈ R and s → 0+ , ln(L(s))/ln s → 0 and ln(sρ L(s))/ln s → ρ.
Proposition 2.7. If g1 ∈ RV Zρ1 , g2 ∈ RV Zρ2 with lims→0 g2 (s) = 0, then g1 ◦g2 ∈
RV Zρ1 ρ2 .
Proposition 2.8 (Asymptotic behavior). If a function L is slowly varying at zero,
then for η > 0 and t → 0+ ,
Rt
(i) 0 sρ L(s)ds ∼
= (1 + ρ)−1 t1+ρ L(t), for ρ > −1;
Rη ρ
(ii) t s L(s)ds ∼
= (−ρ − 1)−1 t1+ρ L(t), for ρ < −1.
Proposition 2.9. Let g ∈ C 1 (0, η] be positive and
lim+
s→0
sg 0 (s)
= +∞.
g(s)
Then g is rapidly varying to zero at zero.
Proposition 2.10. Let g ∈ C 1 (0, η) be positive and
lim+
s→0
sg 0 (s)
= −∞.
g(s)
Then g is rapidly varying to infinity at zero.
Proposition 2.11 ([46, Lemma 2.3]). Let L̂ be defined on (0, η] and be normalized
slowly varying at zero. Then
lim+ R η
t→0
If further
Rη
0
L(τ )
τ dτ
t
L(t)
L(τ )
τ dτ
= 0.
converges, then
lim R t
t→0+
0
L(t)
L(τ )
τ dτ
= 0.
Our results in the section are summarized in the following lemmas.
Lemma 2.12. Let θ ∈ Λ.
(i) limt→0+ Θ(t)
θ(t) = 0;
0
(t)
Θ(t)
d
(ii) limt→0+ Θ(t)θ
= 1 − limt→0+ dt
= 1 − Cθ , and Cθ ∈ [0, 1] when θ
θ 2 (t)
θ(t)
is non-decreasing, Cθ ≥ 1 provided θ is non-increasing;
(iii) when Cθ > 0, θ ∈ N RV Z(1−Cθ )/Cθ and Θ ∈ N RV Z1/Cθ .
Proof. For an arbitrary θ ∈ Λ, we have:
(i) When θ is non-decreasing, we have that 0 < Θ(t) ≤ tθ(t), for all t ∈ (0, δ0 )
and (i) holds; when θ is non-increasing, it follows by θ ∈ L1 (0, δ0 ) that
lim
t→0+
1
Θ(t)
= lim
lim Θ(t) = 0.
θ(t)
t→0+ θ(t) t→0+
(ii) Since
lim
t→0+
Θ(t)θ0 (t)
d Θ(t) = 1 − lim
= 1 − Cθ ,
θ2 (t)
t→0+ dt θ(t)
(2.8)
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BOUNDARY BEHAVIOR OF SOLUTIONS
7
it follows that Cθ ∈ [0, 1] when θ is non-decreasing, and Cθ ≥ 1 provided θ is
non-increasing;
(iii) (1.5) and the l’Hospital’s rule imply
lim
t→0+
Θ(t)
= lim
tθ(t) t→0+
Θ(t)
θ(t)
t
= lim
t→0+
d Θ(t) = Cθ .
dt θ(t)
(2.9)
So, when Cθ > 0, Θ ∈ N RV ZC −1 and it follows by (2.8) and (2.9) that
θ
0
tθ (t)
Θ(t)θ0 (t)
tθ(t)
1 − Cθ
= lim
lim
=
,
t→0 θ(t)
t→0
θ2 (t) t→0 Θ(t)
Cθ
lim
i.e., θ ∈ N RV Z(1−Cθ )/Cθ .
(2.10)
Lemma 2.13 ([51, Lemma 2.2]). Let g satisfy (G1), (G2).
(i) If g satisfies (G3), then Cg ≤ 1;
(ii) (G3) holds with Cg ∈ (0, 1) if and only if g ∈ N RV Z−Cg /(1−Cg ) ;
(iii) (G3) holds with Cg = 0 if and only if g is normalized slowly varying at zero;
(iv) if (G3) holds with Cg = 1, then g is rapidly varying to infinity at zero.
Lemma 2.14 ([51, Lemma 2.3]). Let g satisfy (G1), (G2) and let ψ be uniquely
determined by
Z ψ(t)
dτ
= t, t ∈ [0, ∞).
g(τ
)
0
Then
(i) ψ 0 (t) = g(ψ(t)), ψ(t) > 0, t > 0, ψ(0) = 0 and ψ 00 (t) = g(ψ(t))g 0 (ψ(t)),
t > 0;
(ii) limt→0+ tg(ψ(t)) = 0 and limt→0+ tg 0 (ψ(t)) = −Cg ;
(iii) ψ ∈ N RV Z1−Cg and ψ 0 ∈ N RV Z−Cg ;
(iv) when Cθ + 2Cg > 2 and θ ∈ Λ, limt→0+ ψ(ξΘt2 (t)) = 0 uniformly for ξ ∈
[c1 , c2 ] with 0 < c1 < c2 , where Θ is given as in (1.5);
(v) limt→0+ ψ(ξht1 (t)) = 0 uniformly for ξ ∈ [c1 , c2 ] with 0 < c1 < c2 , where h1
is given in (1.11).
Lemma 2.15. Let q ∈ (0, 1). If Cθ + 2Cg > 2, then
lim (g(ψ(Θ2 (t))))q−1
s→0+
(Θ(t))q
= 0,
(θ(t))2−q
lim g(ψ(Θ2 (t)))θ2 (t) = ∞.
s→0+
Proof. Using Proposition 2.7, Lemma 2.13 (iii) and Lemma 2.15 (iii), we see that
g(ψ(Θ2 (t)))θ2 (t) belongs to N RV Zρ1 with
ρ1 =
−2Cg
2(1 − Cθ )
Cθ + 2Cg − 2 + Cθ
+
=−
< 0,
Cθ
Cθ
Cθ
q
(Θ(t))
and (g(ψ(Θ2 (t))))q−1 (θ(t))
2−q belongs to N RV Zρ2 with
q
2Cg (q − 1) (2 − q)(1 − Cθ )
−
−
Cθ
Cθ
Cθ
Cθ + 2Cg − 2 + Cθ (1 − q) + 2q(1 − Cg )
=
> 0.
Cθ
ρ2 =
Thus the results follow by Proposition 2.6 (ii).
8
B. LI, Z. ZHANG
EJDE-2015/19
3. Local comparison principles
In this section we give some comparison principles near the boundary. For any
δ > 0, we define
Ωδ := {x ∈ Ω : d(x) < δ},
Γδ := {x ∈ Ω : d(x) = δ}.
2
Since ∂Ω ∈ C , there exists a constant δ ∈ (0, min{s0 , δ0 }) which only depends on
Ω such that (see, [22, Lemmas 14.16 and 14.17])
d ∈ C 2 (Ωδ ),
|∇d(x)| = 1,
∆d(x) = −(N − 1)H(x̄) + o(1),
∀x ∈ Ωδ ,
(3.1)
where x̄ is the nearest point to x on ∂Ω, and H(x̄) denotes the mean curvature of
∂Ω at x̄.
Next let v0 ∈ C 2+α (Ω) ∩ C 1 (Ω̄) be the unique solution of the problem
− ∆v = 1,
v > 0,
x ∈ Ω,
v|∂Ω = 0.
(3.2)
By the Höpf maximum principle in [22], we see that
∇v0 (x) 6= 0,
∀x ∈ ∂Ω
and
c1 d(x) ≤ v0 (x) ≤ c2 d(x),
∀x ∈ Ω,
(3.3)
where c1 , c2 are positive constants. We have the lower bound estimations near the
boundary of solutions to (1.1).
Lemma 3.1 (A local comparison principle). For fixed λ > 0, let q ∈ (0, 2], g satisfy
(G1), (G2), b satisfy (B1), and let uλ ∈ C 2 (Ω) ∩ C(Ω̄) be an arbitrary solution to
(1.1) and u0 ∈ C 2 (Ω) ∩ C(Ω̄) be an arbitrary solution to the problem
− ∆u = b(x)g(u),
u > 0,
x ∈ Ω,
u|∂Ω = 0.
(3.4)
Then there exists a positive constant M0 such that
u0 (x) ≤ uλ (x) + M0 v0 (x),
x ∈ Ωδ ,
(3.5)
where δ > 0 sufficiently small such that
u0 (x),
uλ (x) ∈ (0, s0 ),
x ∈ Ωδ ,
where s0 is given as in (G2).
Proof. First, by uλ (x) = v0 (x) = u0 (x) = 0, for all x ∈ ∂Ω, and
u0 , v0 , uλ ∈ C 2 (Ω) ∩ C(Ω̄),
(3.6)
we can choose a large M0 such that
u0 (x) ≤ uλ (x) + M0 v0 (x),
x ∈ Γδ .
Now we prove (3.5). Assume the contrary, there exists x0 ∈ Ωδ such that
u0 (x0 ) − (uλ (x0 ) + M0 v0 (x0 )) > 0.
It follows that there exists x1 ∈ Ωδ such that
0 < u0 (x1 ) − (uλ (x1 ) + M0 v0 (x1 )) = max (u0 (x) − (uλ (x) + M0 v0 (x))).
x∈Ω̄δ
Then ([22, Theorem 2.2])
∆(u0 − (uλ + M0 v0 ))(x1 ) ≤ 0.
On the other hand, we see by (B1), (G1) and (G2) that
∆(u0 − (uλ + M0 v0 ))(x1 )
= −∆uλ (x1 ) + M0 + ∆u0 (x1 )
(3.7)
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BOUNDARY BEHAVIOR OF SOLUTIONS
9
= b(x1 )(g(uλ (x1 )) − g(u0 (x1 ))) + M0 + λ|∇uλ (x1 )|q + σ > 0,
which is a contradiction. Hence (3.5) holds.
Next we consider the upper bound estimations near the boundary to uλ . For
q ∈ (0, 1), we have the following lemma.
Lemma 3.2 (A local comparison principle). For fixed λ > 0, let g satisfy (G1),
(G2), b satisfy (B1), and let uλ ∈ C 2 (Ω) ∩ C(Ω̄) be an arbitrary solution to (1.1),
ūλ ∈ C 2 (Ωδ ) ∩ C(Ω̄δ ) satisfy
− ∆ūλ ≥ b(x)g(ūλ ) + λ|∇ūλ |q + σ,
ūλ > 0,
x ∈ Ωδ ,
ūλ |∂Ω = 0,
(3.8)
where δ > 0 sufficiently small such that
ūλ (x),
uλ (x) ∈ (0, s0 ),
x ∈ Ωδ ,
where s0 is given as in (G2). Then there exists a positive constant M0 such that
uλ (x) ≤ ūλ (x) + λM0 v0 (x),
x ∈ Ωδ .
(3.9)
Proof. From uλ (x) = ūλ (x) = v0 (x) = 0, for all x ∈ ∂Ω, and
uλ ∈ C 2 (Ω) ∩ C(Ω̄), v0 ∈ C 2 (Ω) ∩ C 1 (Ω̄), ūλ ∈ C 2 (Ωδ ) ∩ C(Ω̄δ ),
(3.10)
we can choose a large M0 such that
uλ (x) ≤ ūλ (x) + λM0 v0 (x),
M01−q
q
x ∈ Γδ ,
(3.11)
q
(3.12)
≥ λ max |∇v0 (x)| .
x∈Ω̄
Now we prove (3.9). Assume the contrary, there exists x0 ∈ Ωδ such that
uλ (x0 ) − (ūλ (x0 ) + λM0 v0 (x0 )) > 0.
It follows that there exists x1 ∈ Ωδ such that
0 < uλ (x1 ) − (ūλ (x1 ) + λM0 v0 (x1 )) = max (uλ (x) − (ūλ (x) + λM0 v0 (x))).
x∈Ω̄δ
Then ([22, Theorem 2.2])
∇(uλ − (ūλ + λM0 v0 ))(x1 ) = 0
and
∆(uλ − (ūλ + λM0 v0 ))(x1 ) ≤ 0.
On the other hand, using the basic inequality for q ∈ (0, 1)
|sq2 − sq1 | ≤ |s2 − s1 |q ,
∀s2 , s1 ≥ 0,
it follows by (B1), (G1) and (G2) that
∆ uλ − (ūλ + λM0 v0 ) (x1 )
= −∆ūλ (x1 ) + λM0 + ∆uλ (x1 )
≥ b(x1 )(g(ūλ (x1 )) − g(uλ (x1 ))) + λ(M0 + |∇ūλ (x1 )|q − |∇uλ (x1 )|q )
> λ(M0 − λq M0q |∇v0 (x1 )|q ) > 0,
which is a contradiction. Hence (3.9) holds.
For q ∈ [1, 2] and an arbitrary positive constant C, by using the following inequality [47, (3.10)]
s2
sq ≤ 1−q/2 + C q/2 , ∀s ≥ 0,
C
we see that
−∆uλ ≤ b(x)g(uλ )+λC q/2−1 |∇uλ |2 +λC q/2 +σ, uλ > 0, x ∈ Ω, uλ |∂Ω = 0. (3.13)
10
B. LI, Z. ZHANG
EJDE-2015/19
We can choose C such that the problem
−∆ūλ = b(x)g(ūλ ) + λC q/2−1 |∇ūλ |2 + λC q/2 + σ, ūλ > 0, x ∈ Ω,
ūλ |∂Ω = 0,
(3.14)
has one classical solution ūλ ([48, Theorem 4.1]).
For a fixed λ, let uλ and ūλ be arbitrary solutions to (3.13) and (3.14), we see
that the nonlinear changes of variable
wλ = exp(ηuλ ) − 1 and w̄λ = exp(ηūλ ) − 1
transform problems (3.13) and (3.14) into the equivalent problems
− ∆wλ ≤ b(x)g̃(wλ ) + ηf (wλ ), wλ > 0,
x ∈ Ω,
wλ |∂Ω = 0,
(3.15)
and
− ∆w̄λ = b(x)g̃(w̄λ ) + ηf (w̄λ ),
w̄λ > 0,
x ∈ Ω,
w̄λ |∂Ω = 0,
(3.16)
respectively. Where
g̃(s) = η(1 + s)g(η −1 ln(1 + s)),
η = λC q/2−1 ,
f (s) = (ηC + σ)(1 + s).
(3.17)
(3.18)
Lemma 3.3. For fixed λ > 0. Let g satisfy (G1)–(G3). Then
(i) g̃ ∈ C 1 ((0, ∞), (0, ∞)) and lims→0 g̃(s) = ∞;
(ii) when one of the following conditions holds
(S01) Cg > 0;
(S02) Cg = 0 and λ lim sups→0+ |gg(s)
0 (s)| < 1,
0
there exists s1 > 0 such that g̃ (s) < 0, ∀s ∈ (0, s1 );
(iii)
Z s
dτ
= −Cg .
lim+ g̃ 0 (s)
s→0
0 g̃(τ )
Proof. By (G1), (i) is obvious. (ii) follows by [50, Lemma 3.1]. (iii) Since g satisfies
(G1) and is decreasing on (0, s0 ), we see that
Z s
dτ
s
0<
<
, ∀s ∈ (0, s0 ),
g(τ
)
g(s)
0
i.e.,
Z s
dτ
0 < g(s)
< s, ∀s ∈ (0, s0 ),
(3.19)
g(τ
)
0
Z s
dτ
lim g(s)
= 0.
(3.20)
g(τ
)
s→0+
0
Let υ = η −1 ln(1 + τ ) and ς = η −1 ln(1 + s). It follows by (3.20) and (G3) that
Z s
dτ
0
lim g̃ (s)
+
s→0
0 g̃(τ )
Z
s
dτ
0 −1
−1
= lim+ g (η ln(1 + s)) + ηg(η ln(1 + s))
−1 ln(1 + τ ))
η(1
+
τ
)g(η
s→0
0
Z ς
Z ς
dυ
dυ = lim+ g 0 (ς)
+ ηg(ς)
= −Cg .
g(υ)
g(υ)
ς→0
0
0
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BOUNDARY BEHAVIOR OF SOLUTIONS
11
Thus we have the following comparison principle.
Lemma 3.4 ([50, Lemma 3.1]). For fixed λ > 0, let f ∈ C([0, ∞), [0, ∞)), g satisfy
(G1), (G2), b satisfy (B1). Then there exists a positive constant M0 such that
wλ (x) ≤ w̄λ (x) + M0 (ηC + σ)v0 (x),
x ∈ Ωδ ,
(3.21)
where δ > 0 sufficiently small such that
wλ (x), w̄λ (x) ∈ (0, s1 ),
x ∈ Ωδ ,
where s1 is as in Lemma 3.3.
4. Boundary behavior
In this section we prove Theorems 1.1–1.3. First we have the statement in [30,
Theorem 1.1] with a ≡ 1 in Ω.
Lemma 4.1. For a fixed λ > 0, let f ∈ C([0, ∞), [0, ∞)), g satisfy (G1)–(G3), and
let b satisfy (B1), (B2). If
Cθ + 2Cg > 2,
(4.1)
then for any classical solution Vλ to the problem
− ∆V = b(x)g(V ) + λa(x)f (V ),
V |∂Ω = 0,
(4.2)
Vλ (x)
Vλ (x)
1−C
≤ lim sup
≤ ξ2 g ,
ψ(Θ2 (d(x))) d(x)→0
ψ(Θ2 (d(x)))
(4.3)
V > 0,
x ∈ Ω,
it holds that
1−Cg
ξ1
≤ lim inf
d(x)→0
where ψ is the solution to (1.2), ξ1 and ξ2 are given as in (1.8). In particular, (i)
and (ii) in Theorem 1.1 hold.
Next we have the statement in [51, Theorems 1.2] with a ≡ 1 in Ω.
Lemma 4.2. For a fixed λ > 0, let f ∈ C([0, ∞), [0, ∞)), g satisfy (G1)–(G3), and
let b satisfy (B1). If b satisfies (B3), then any classical solution Vλ to (4.2) satisfies
(1.10).
Next we have the statement in [51, Theorems 1.3] with a ≡ b in Ω.
Lemma 4.3. For a fixed λ > 0, let f ∈ C([0, ∞), [0, ∞)), g satisfy (G1) and
g(s) = s−γ + µsp , s ∈ (0, s0 ) for some s0 > 0 and γ, p, µ > 0, and let b satisfy (B1).
If b satisfies (B4), then any classical solution Vλ to (4.2) satisfies (1.12).
Remark 4.4. Obviously, when f ≡ 0 on [0, ∞), a solution Vλ to (4.2) is a solution
to (3.4).
Proof of Theorem 1.1. Let uλ ∈ C 2 (Ω) ∩ C(Ω̄) be an arbitrary solution to (1.1).
Using (3.3), Lemmas 3.1, 3.3, 3.4, 4.1 and 2.14 (iv), we obtain that for q ∈ (0, 2],
1−Cg
ξ1
≤ lim inf
d(x)→0
u0 (x)
uλ (x)
≤ lim inf
,
ψ(Θ2 (d(x))) d(x)→0
ψ(Θ2 (d(x)))
(4.4)
and for q ∈ [1, 2],
lim sup
d(x)→0
wλ (x)
w̄λ (x)
1−C
≤ lim sup
≤ ξ2 g .
ψ1 (Θ2 (d(x))) d(x)→0
ψ1 (Θ2 (d(x)))
(4.5)
12
B. LI, Z. ZHANG
EJDE-2015/19
where wλ (x) = exp(ηuλ (x)) − 1, w̄λ (x) = exp(ηūλ (x)) − 1, η = λC q/2−1 , ψ1 is the
solution to the problem
Z ψ1 (t)
ds
= t, ∀t > 0,
(4.6)
g̃(s)
0
and g̃ is given in (3.17).
From
ψ(t) = η −1 ln(1 + ψ1 (t)), ∀t > 0,
exp(ηs) − 1 ∼
= ηs as s → 0,
it follows that q ∈ [1, 2],
lim sup
d(x)→0
uλ (x)
1−C
≤ ξ2 g .
2
ψ(Θ (d(x)))
(4.7)
Thus (1.7) holds for q ∈ [1, 2].
Next we structure an appropriate supersolution near the boundary to (1.1) in
the case q ∈ (0, 1). Let ε ∈ (0, b1 /4) and let
τ1 = ξ2 + 2εξ2 /b2 ,
where ξ2 is given in (1.8). It follows that ξ2 < τ1 < 2ξ2 , limε→0 τ1 = ξ2 , and
− 4τ1 Cg + 2τ1 (2 − Cθ ) + b2 = −2ε.
(4.8)
By (B2), (3.1), Lemmas 2.12, 2.14 and 2.15, we see that
lim τ1 Θ2 (d(x))g 0 (ψ(τ1 Θ2 (d(x)))) = −Cg ,
d(x)→0
θ0 (d(x))Θ(d(x))
Θ(d(x))
∆d(x) = 2 − Cθ ,
θ(d(x))
d(x)→0
(Θ(d(x)))q
lim λτ1q 2q
2−q
(θ(d(x))) (g(ψ(τ1 Θ2 (d(x)))))1−q
d(x)→0
σ
+ 2
= 0,
θ (d(x))g(ψ(τ1 Θ2 (d(x))))
b(x)
lim sup 2
≤ b2 .
θ
(d(x))
d(x)→0
lim
θ2 (d(x))
+1+
Thus, corresponding to ε, s0 and δ, where s0 is given in (G2) and δ in Lemma 3.1,
respectively, there is δε ∈ (0, δ) sufficiently small such that for x ∈ Ωδε ,
ūε = ψ(τ1 Θ2 (d(x)))
satisfies
ūε (x) ∈ (0, s0 ),
x ∈ Ωδε ,
(4.9)
and
∆ūε (x) + b(x)g(ūε (x)) + λ|ūε (x)|q + σ
= ψ 00 (τ1 Θ2 (d(x)))(2τ1 Θ(d(x))θ(d(x)))2 + 2τ1 ψ 0 (τ1 Θ2 (d(x)))
× θ2 (d(x)) + Θ(d(x))θ0 (d(x)) + Θ(d(x))θ(d(x))∆d(x)
+ b(x)g(ψ(τ1 Θ2 (d(x)))) + λ(2τ1 )q (θ(d(x))Θ(d(x)))q (g(ψ(τ1 Θ2 (d(x)))))q + σ
= g(ψ(τ1 Θ2 (d(x))))θ2 (d(x)) 4τ1 τ1 Θ2 (d(x))g 0 (ψ(τ1 Θ2 (d(x))))
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BOUNDARY BEHAVIOR OF SOLUTIONS
13
θ0 (d(x))Θ(d(x))
Θ(d(x))
b(x)
+1+
∆d(x) + 2
2
θ (d(x))
θ(d(x))
θ (d(x))
q
(Θ(d(x)))
σ
+ λτ1q 2q
+
(θ(d(x)))2−q (g(ψ(τ1 Θ2 (d(x)))))1−q
θ2 (d(x))g(ψ(τ1 Θ2 (d(x))))
≤ 0;
+ 2τ1
i.e., ūε is a supersolution of equation (1.1) in Ωδε .
Let uλ ∈ C(Ω̄) ∩ C 2+α (Ω) be an arbitrary classical solution to (1.1). By Lemma
3.2, we see that there exists M0 > 0 such that for x ∈ Ωδε ,
uλ (x) ≤ ūε (x) + λM0 v0 (x),
i.e.,
v0 (x)
uλ (x)
≤ 1 + λM0
,
2
ψ(τ1 Θ (d(x)))
ψ(τ1 Θ2 (d(x)))
It follows by (3.3) and Lemma 2.14 (iv) that
lim sup
d(x)→0
x ∈ Ωδε .
uλ (x)
≤ 1.
ψ(τ2 Θ2 (d(x)))
Using Lemma 2.14 again, we have
ψ(τ1 Θ2 (d(x)))
= τ1 1−Cg .
d(x)→0 ψ(Θ2 (d(x)))
lim
Moreover, since Cθ > 0, by (2.9) and Lemma 2.14, we obtain that
ψ(Θ2 (d(x)))
2(1−Cg )
= Cθ
.
2
2
ψ(d
(x)θ
(d(x)))
d(x)→0
lim
Thus letting ε → 0, we have
lim sup
d(x)→0
uλ (x)
1−C
≤ ξ2 g .
ψ(Θ2 (d(x)))
(4.10)
Combining (4.10) with (4.4), we obtain (1.7). In particular, when Cg = 1, uλ
satisfies
uλ (x)
= 1;
lim
d(x)→0 ψ(Θ2 (d(x)))
and, when Cg < 1 and b1 = b2 = b0 in (b1), uλ satisfies
uλ (x)
2
d(x)→0 ψ(d (x)θ 2 (d(x)))
lim
= (ξ01 Cθ2 )1−Cg .
This completes the proof of Theorem 1.1.
Proof of Theorem 1.2. Let uλ ∈ C 2 (Ω) ∩ C(Ω̄) be an arbitrary solution to (1.1).
For q ∈ [1, 2], in a similar way as that of Theorem 1.1, by using (3.1), (3.3), Lemmas
3.1, 3.3, 3.4, 4.2 and 2.14 (v), we can show that Theorem 1.2 holds.
Next we construct an appropriate supersolution near the boundary to (1.1) in
the case of q ∈ (0, 1). Let ε ∈ (0, b1 /4) and let τ2 = b2 + 2ε. It follows that
b1 /2 < τ2 < 2b2 .
By (B3), (G1), (3.1), (3.3), Propositions 2.6 (iii) and 2.11, and Lemma 2.14, we
derive that
d2 (x)
σ lim
= 0,
d(x)→0 L̂(d(x))g(ψ(τ2 h1 (d(x))))
14
B. LI, Z. ZHANG
λτ2q lim
d(x)→0
EJDE-2015/19
(d(x))2−q (L̂(d(x)))q−1 (g(ψ(τ2 h1 (d(x)))))1−q = 0,
lim τ2 h1 (d(x))g 0 (ψ(τ2 h1 (d(x)))) = −Cg ,
d(x)→0
L̂(d(x))
= 0,
d(x)→0 h1 (d(x))
lim
lim τ2 1 −
d(x)→0
d(x)L̂0 (d(x)) L̂(d(x))
b(x)
lim sup
d(x)→0 d−2 (x)L̂(d(x))
= τ2 ,
≤ b2 ,
lim τ2 d(x)∆d(x) = 0.
d(x)→0
Thus, corresponding to ε, s0 and δ, where s0 is given as in (G2) and δ in Lemma
3.1, respectively, there is δε ∈ (0, δ) sufficiently small such that for x ∈ Ωδε
ūε = ψ(τ2 h1 (d(x)))
satisfies (4.9) and
∆ūε (x) + b(x)g(ūε (x)) + λ|ūε (x)|q + σ
= ψ 00 (τ2 h1 (d(x)))τ22 h21 (d(x)) + ψ 0 (τ2 h1 (d(x))) τ2 h001 (d(x)) + τ22 h01 (d(x))∆d(x)
+ b(x)g(ψ(τ2 h(d(x)))) + λτ q
L̂(d(x)) q
(g(ψ(τ2 h1 (d(x)))))q + σ
d(x)
= (d(x))−2 L̂(d(x))g(ψ(τ2 h1 (d(x))))
L̂(d(x))
d(x)L̂0 (d(x)) × τ2 τ2 h1 (d(x))g 0 (ψ(τ2 h1 (d(x))))
− τ2 1 −
h1 (d(x))
L̂(d(x))
b(x)
d2 (x)
1
+ τ2 d(x)∆d(x) +
+σ
d−2 (x)L̂(d(x))
L̂(d(x)) g(ψ(τ2 h1 (d(x))))
1
+ λτ2q (d(x))2−q (L̂(d(x)))q−1
(g(ψ(τ2 h1 (d(x)))))1−q
≤ 0,
i.e., ūε is a supersolution to equation (1.1) in Ωδε .
The rest of the proof is the same as that Theorem 1.1 and is omitted.
Proof of Theorem 1.3. Let uλ ∈ C 2 (Ω) ∩ C(Ω̄) be an arbitrary solution to (1.1).
For q ∈ [1, 2], in a similar way as that of Theorem 1.1, by using (3.1), (3.3),
Lemmas 3.1, 3.3, 3.4 and 4.3, we can show that Theorem 1.3 holds.
Next we construct an appropriate supersolution near the boundary to (1.1) in
the case of q ∈ (0, 1). Let ε ∈ (0, 1). Let τ3 be the unique positive solution to the
problem
t
b2 t−γ −
= −2ε,
1+γ
it follows by the properties of the function bi t−γ −
(b1 (1 + γ))1/(1+γ) < τ3 < ζ0 ,
t
1+γ
lim τ3 = (b2 (1 + γ))1/(1+γ) ,
ε→0
where ζ0 is the unique positive solution to the problem
b2 t−γ −
(i = 1, 2) that
t
= −2.
1+γ
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BOUNDARY BEHAVIOR OF SOLUTIONS
15
Rη
)
Since L̂ and t L̂(τ
τ dτ are slowly varying at zero, we see that by (3.3), (B4),
Propositions 2.6 and 2.11 that
Z
d(x) η L̂(τ )
1
lim
dτ −
L̂(d(x)) ∆d(x) = 0,
τ
1+γ
d(x)→0 L̂(d(x))
d(x)
1 d(x)L̂0 (d(x))
= 0,
d(x)→0 1 + γ
L̂(d(x))
L̂(d(x))
γ
= 0,
R
2
η
L̂(τ )
d(x)→0 (1 + γ)
dτ
d(x) τ
lim
lim
lim sup
d(x)→0
b(x)
(d(x))γ−1 L̂(d(x))
≤ b2 ,
and, using (G1) and (B4), there holds
Z
d(x) η
L̂(τ ) 1/(1+γ)
dτ
= 0;
τ
d(x)→0 L̂(d(x))
d(x)
Z η L̂(τ ) (γ+p)/(1+γ)
p−1
γ+p
= 0;
µτ3
lim (d(x))
dτ
τ
d(x)→0
d(x)
Z η L̂(τ ) (q+γ)/(1+γ)
λτ3q−1 lim d(x)
dτ
τ
d(x)→0
d(x)
Z η L̂(τ ) −1 q 1
× 1 −
dτ
L̂(d(x))
= 0.
1+γ
τ
d(x)
στ3−1
lim
Thus, corresponding to ε, s0 and δ, where s0 is given as in (G2) and δ in Lemma
3.1, respectively, there is δε ∈ (0, δ) sufficiently small such that for x ∈ Ωδε
Z η L̂(τ ) 1/(1+γ)
ūε = τ3 d(x)
dτ
τ
d(x)
satisfies (4.2) and
∆ūε (x) + b(x)((ūε (x))−γ + µ(ūε (x))p ) + λ|ūε (x)|q + σ
Z
L̂(d(x)) η L̂(τ ) −γ/(1+γ)
γ
L̂(d(x))
1
= τ3
dτ
−
−
2 Rη
L̂(τ )
d(x)
τ
1
+
γ
(1
+
γ)
d(x)
dτ
d(x)
0
Z
d(x) η
τ
1 d(x)L̂ (d(x))
L̂(τ )
1
+
dτ −
L̂(d(x)) ∆d(x)
1 + γ L̂(d(x))
1+γ
L̂(d(x)) d(x) τ
Z η L̂(τ ) (γ+p)/(1+γ) b(x)
dτ
+
τ3−γ−1 + µτ3p−1 (d(x))γ+p
τ
(d(x))γ−1 L̂(d(x))
d(x)
Z η L̂(τ ) γ/(1+γ)
−1 d(x)
+ στ3
dτ
L̂(d(x)) d(x) τ
Z η L̂(τ ) (q+γ)/(1+γ) Z η L̂(τ ) −1 q 1
q−1
+ λτ3 d(x)
dτ
L̂(d(x))
dτ
1 −
τ
1
+
γ
τ
d(x)
d(x)
≤ 0,
−
i.e., ūε is a supersolution of (1.1) in Ωδε . The conclusion follows as in the proof of
Theorem 1.1.
16
B. LI, Z. ZHANG
EJDE-2015/19
4.1. Acknowledgments. This work is supported in part by NNSF of China under
grant 11301301.
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Bo Li
School of mathematics and statistics, Lanzhou University, Lanzhou 730000, Gansu,
China.
School of Mathematics and Information Science, Yantai University, Yantai 264005,
Shandong, China
E-mail address: libo yt@163.com
Zhijun Zhang
School of Mathematics and Information Science, Yantai University, Yantai 264005,
Shandong, China
E-mail address: chinazjzhang2002@163.com, zhangzj@ytu.edu.cn