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Large solutions with a power nonlinearity given by a variable exponent for p -Laplacian equations
Nonlinear Analysis 110 (2014) 130–140
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Large solutions with a power nonlinearity given by a variable exponent for p-Laplacian equations✩ Yujuan Chen ∗ , Yueping Zhu, Ruiya Hao School of Science, Nantong University, Nantong 226007, PR China
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abstract
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Article history: Received 19 April 2014 Accepted 4 August 2014 Communicated by Enzo Mitidieri
In this paper we establish the existence, uniqueness and blow-up rate near the boundary of boundary blow-up solutions to p-Laplacian elliptic equations with a power nonlinearity given by a variable exponent −∆p u = b(x)uq(x) , where ∆p u = div(|∇ u|p−2 ∇ u) with p > 1. Here we combine the localization method originated by López-Gómez (2003) with some previous radially symmetric results to get the asymptotic behavior near the boundary. © 2014 Elsevier Ltd. All rights reserved.
MSC: 35B50 35J65 35B32 Keywords: p-Laplacian equations A variable exponent Boundary blow-up solutions Existence and uniqueness Blow-up rate
1. Introduction and main results In this paper, we will consider the following boundary blow-up problem for p-Laplacian elliptic equations
∆p u = b(x)uq(x) , u = ∞,
x ∈ Ω, x ∈ ∂Ω,
(1.1)
¯ ) will where Ω ⊂ RN (N ≥ 2) is a smooth bounded domain, ∆p u = div(|∇ u|p−2 ∇ u) with p > 1, the exponent q(x) ∈ C η (Ω ¯ ) for some 0 < λ < 1 with b(x) ≥ 0 and b(x) ̸≡ 0 in Ω . Set be a positive continuous function, and b(x) ∈ C λ (Ω Ω0 = int {x ∈ Ω : b(x) = 0}
¯ 0 ⊂ Ω and b(x) > 0 in Ω \ Ω ¯ 0. and suppose, throughout this paper, that Ω 1 ,p 1 By a solution to (1.1) we mean a positive function u ∈ Wloc (Ω ) ∩ C (Ω ) such that (1.1) holds in the distribution sense, and u(x) → ∞ as x ∈ Ω and d(x) → 0, where d(x) = dist(x, ∂ Ω ) represents the distance from x to ∂ Ω for x ∈ Ω . Such a solution of (1.1) is called a boundary blow-up (or large) solution. When q is a constant, the problem (1.1) has been studied by many authors. We quote the papers [1–12] and the references therein for semilinear problems and, [13–17] etc. for problems with the p-Laplacian. However, the main precursor of the
✩ This work was supported by PRC Grants NSFC 11271209, 11371370 and Jiangsu Education Commission 13KJB110023.
∗
Corresponding author. Tel.: +86 0513 85015885. E-mail address: [email protected] (Y. Chen).
http://dx.doi.org/10.1016/j.na.2014.08.003 0362-546X/© 2014 Elsevier Ltd. All rights reserved.
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
131
differential equation of (1.1) in the available mathematical literature is the porous media equation
∆v m(x) = b(x)v, v = 0,
x ∈ Ω, x ∈ ∂Ω,
with variable m(x), which goes back to [18,19]. But, at the best of our knowledge, the previous work concerning large solutions with nonlinearities given a variable exponent is much less. When large solution is concerned, the pioneering paper ¯ and q > 1 in Ω , q = 1 on ∂ Ω , was considered, and the exis [20] where the problem ∆u = −λu + a(x)uq(x) with a > 0 in Ω istence of a maximal and a minimal positive solution was obtained. Then J. García-Melián et al. [21,22] studied the existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem ∆u = uq(x) and ∆u = eq(x)u respectively. As we know, when q is a constant, the necessary and sufficient condition for the existence of the solution to (1.1) is q > p − 1. In our work, however, the function q(x) may be less than p − 1. In addition to considering the existence and nonexistence of positive solutions to (1.1), we also consider the uniqueness and the blow-up rate of solutions near the boundary of the domain. Even though most of the proofs are adaptations of the corresponding ones in [21], we still need to overcome some difficulties. Firstly, sometimes it may not be possible to extend results from p = 2 to p ̸= 2, since many nice features inherent to p = 2 are lost or difficult to verify once p ̸= 2. Secondly, we must face the difficulties caused by the nonnegative weighted function b(x) in the nonlinear term. Therefore, we extend the results in [21,20] to some extent. To prove the existence of the solution to (1.1), we use techniques mainly based on comparison, referring to problem (1.1) with q as a constant. To prove asymptotic behavior near the boundary, we combine the localization method originated by [8] with some radially symmetric results. Now we will state our results. We firstly show that positive solutions to (1.1) are possible only if q(x) ≥ p − 1 on ∂ Ω . Theorem 1.1. Assume that there exists x0 ∈ ∂ Ω such that q(x0 ) < p − 1. Then the problem (1.1) has no positive solutions. ¯. Moreover, the same conclusion holds if q(x) ≤ p − 1 in a whole neighborhood of x0 ∈ ∂ Ω relative to Ω Thanks to Theorem 1.1, we always need q(x) ≥ p − 1 on ∂ Ω in order to have positive solutions. We will make the assumption that q(x) > p − 1 in a neighborhood of ∂ Ω , although q may be p − 1 on ∂ Ω . We also remark that q(x) ≤ p − 1 is permitted at interior points, and we can still get a solution. Theorem 1.2. Assume that q(x) > p−1 in the strip Ωδ = {x ∈ Ω : dist(x, ∂ Ω ) < δ} for some δ > 0. Then problem (1.1) admits at least a positive solution. The following two theorems give the boundary blow-up behavior of positive solutions to (1.1) in the case, where q(x) > p − 1 and q(x) = p − 1 respectively somewhere on ∂ Ω . Subsequently, we denote by n : ∂ Ω → RN , x → n(x) := nx , the outward unit normal vector-field of Ω , and for each ω ∈ (0, π /2), Cx0 ,ω := x ∈ Ω : angle(x − x0 , −nx0 ) ≤ π /2 − ω .
Theorem 1.3. Assume there exists x0 ∈ ∂ Ω such that q(x0 ) > p − 1. Let u be a positive solution to (1.1).
¯ ) and two positive constants A1 and A2 , such that (i) If in the neighborhood of x0 (relative to Ω ), there exists µ(x) ∈ C η (Ω A1 d(x)µ(x) ≤ b(x) ≤ A2 d(x)µ(x)
(1.2)
then there exist a neighborhood V of x0 (relative to Ω ) and positive constants C1 , C2 such that C1 d(x)−α(x) ≤ u(x) ≤ C2 d(x)−α(x)
in V ,
(1.3)
where
µ(x) + p . q(x) − p + 1 (ii) If there exist κ = κ(x0 ) > 0, µ = µ(x0 ) ≥ 0 such that α(x) =
b(x)
= 1, κ d(x)µ then for each ω ∈ (0, π /2), one has that lim
(1.4)
x→x0
lim
x→x0 ,x∈Cx0 ,ω
u(x) Md(x)α(x)
= 1,
(1.5)
where
µ+p α= , q−p+1
M =
1
κ
(p − 1)(α + 1)α
p−1
q−p1+1
.
(1.6)
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Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
Corollary 1.1. Suppose that q(x) > p − 1 on ∂ Ω , and there exist κ ∈ C (∂ Ω , (0, ∞)), µ ∈ C (∂ Ω , (0, ∞)) such that lim
x→x0
b(x) = 1 uniformly in x0 ∈ ∂ Ω . κ(x0 )d(x)µ(x0 )
(1.7)
Then any positive solution u of (1.1) satisfies lim
x→x0
u( x ) M (x0 )d(x)α(x0 )
uniformly in x0 ∈ ∂ Ω ,
= 1,
where
µ(x0 ) + p , α(x0 ) = q(x0 ) − p + 1
M (x0 ) =
1
κ(x0 )
(p − 1)(α(x0 ) + 1)α(x0 )
Therefore, for any pair u1 , u2 of positive solutions of (1.1), limx→x0
p−1
u1 (x) u2 (x)
1 q(x0 )−p+1
.
(1.8)
= 1 uniformly in x0 ∈ ∂ Ω .
Theorem 1.4. Let x0 ∈ ∂ Ω be a point with q(x0 ) = p − 1. If there exist constants µ ≥ 0, κ, γ , Q > 0 such that lim
x→x0
b(x) = 1, κ d(x)µ
lim
q(x) − p + 1 d(x)γ
x→x0
= Q,
(1.9)
then for every positive solution u of (1.1): lim
d(x)γ log u(x)
x→x0
− log d(x)
=
p + pγ + µ Q
.
(1.10)
It is natural to ask under which conditions the solution provided by Theorem 1.2 is unique. It turns out that q > p − 1 on ∂ Ω is sufficient as long as q ≥ p − 1 in the whole Ω . Theorem 1.5. Under the conditions of Corollary 1.1 and q(x) ≥ p − 1 in Ω , the problem (1.1) admits a unique positive solution. The paper is organized as follows: in Section 2 we prove Theorems 1.1 and 1.2. Section 3 will be dedicated to proving the boundary estimates and uniqueness, which are stated in Theorems 1.3–1.5. 2. Existence and nonexistence In this section, we deal with the existence and nonexistence of positive solutions to problem (1.1). First we cite a comparison result for a class of quasilinear equations. Lemma 2.1 (See [23] for a Proof). Let G : Q × R → R be non-increasing in the second variable and continuous. Let u, v ∈ W 1,p (Q) satisfy the respective inequalities
|∇ u|p−2 ∇ u · ∇ϕ ≤ Q
G(x, u)ϕ, Q
and
|∇v|p−2 ∇v · ∇ϕ ≥ Q
G(x, v)ϕ, Q
1,p
for all non-negative ϕ ∈ W0 (Q). Then the inequality u ≤ v on ∂ Q implies u ≤ v in Q. We now show that there are no solutions if q(x) < p − 1 somewhere on ∂ Ω (alternatively, q(x) ≤ p − 1 in a neighborhood of a boundary point). Throughout the paper, we denote by B(x, r ) the ball of center x and radius r.
¯. Proof of Theorem 1.1 (See the Proof of Theorem 1 in [21] or Theorem 1 in [24]). Let r > 0 such that q(x) < p−1 in B(x0 , 3r )∩Ω Now choose a smooth subdomain D of B(x0 , 3r ) ∩ Ω such that ∂ D ∩ ∂ Ω contains B(x0 , 2r ) ∩ ∂ Ω . Let ψ be a smooth function supported on ∂ D which verifies 0 ≤ ψ ≤ 1, ψ = 1 on B(x0 , r ) ∩ ∂ Ω and ψ = 0 on ∂ D \ (B(x0 , 2r ) ∩ ∂ Ω ). Since u = 0 is a subsolution and u = n is a supersolution, the monotone method in [25] then implies the problem ∆p z = b(x)z q(x) , z = nψ,
x ∈ D, x ∈ ∂D
has a positive solution zn for every positive integer n and since q(x) > 0, by Lemma 2.1, it is unique. Moreover, if (1.1) has a positive solution, it follows by comparison that u ≥ zn
in D,
since u ≥ nψ on ∂ D for every n.
(2.1)
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
133
On the other hand, we have zn = nwn , where wn solves
∆p w = nq(x)−p+1 b(x)w q(x) , z = ψ,
x ∈ D, x ∈ ∂ D.
Now 0 ≤ wn ≤ 1 and q(x) < p − 1, so it is standard to conclude (for a subsequence if necessary) that wn → w0 as n → +∞, where w0 is the solution of the following problem
∆p w0 = 0, w = ψ,
x ∈ D, x ∈ ∂ D.
Since w0 > 0 in D, we obtain that zn → +∞ uniformly in compact subsets of D ∪ (B(x0 , r ) ∩ ∂ Ω ). But then (2.1) implies u = +∞ in D ∪ (B(x0 , r ) ∩ ∂ Ω ), which is not possible. Hence no positive solution to (1.1) exists. Finally, observe that the previous argument continues to be valid – with only minor changes – if q(x) ≤ p − 1 in a neighborhood of a point x0 ∈ ∂ Ω . Thus the proof is concluded. Now we prove the existence. The approach is the standard one: we construct solutions with finite datum on ∂ Ω and then show that they are locally uniformly bounded. Proof of Theorem 1.2. Let n be a positive integer. By a similar proof as in Theorem 1.1, the problem
∆p u = b(x)uq(x) , u = n,
x ∈ Ω, x ∈ ∂Ω
(2.2)
has a unique positive solution which is denoted by un . Thanks to un+1 is a supersolution to (2.2) and by uniqueness un+1 ≥ un . Let us prove next that un is bounded in compact subsets of Ω . Taking δ small, we can assume un > 1 in a strip Ωδ = {x ∈ Ω : dist(x, ∂ Ω ) < δ} for all n. Fix ε with 0 < ε < δ and a point x0 such that d(x0 ) = ε/2. Since q(x) > p − 1 in Ωδ , we q have that q(x) ≥ q0 > p − 1 in B(x0 , ε/4) and thus ∆un ≥ b(x)un0 in B(x0 , ε/4). Hence in B(x0 , ε/4), un ≤ U, the positive solution to
∆p U = b(x)U q0 , u = +∞,
x ∈ B(x0 , ε/4), x ∈ ∂ B(x0 , ε/4).
(2.3)
In fact, the existence of the solution to problem (2.3) can be found in [13, Theorem 1.2]. This shows that un is uniformly bounded in B(x0 , ε/8). A compactness argument proves that un is uniformly bounded in the set {x ∈ Ω : d(x) = ε/2}, and since ∆p un ≥ 0 in Ω , we obtain uniform bounds in the whole {x ∈ Ω : d(x) > ε/2}. Since ε was arbitrarily small, the sequence {un } is locally uniformly bounded in Ω . Finally, standard regularity and compactness arguments show that limn→∞ un (x) = u∞ (x) exists and is a positive blow-up solution of (1.1). This finishes the proof. 3. Blow-up rates and uniqueness Firstly, we give a rough estimate of the solutions to (1.1). The idea of the proof comes from [21]. Proof of Theorem 1.3(i). Choose a neighborhood V ′ of x0 such that q(x) > p − 1 in V ′ , meanwhile, (1.2) holds in V ′ (we can take for instance a ball centered at x0 intersected with Ω ). By diminishing the radius of V ′ , we can select a smaller neighborhood V such that B(x, d(x)/2) ⊂ V ′ for x ∈ V . Take x ∈ V and define the scaled function
v(y) = d(x)α(x) u (x + yd(x)/2) , for y ∈ B := B(0, 1). Applying (1.1)–(1.2), it can be checked that the function v satisfies
∆p v =
= ≥
1 2p 1 2p 1 2p
d(x)(α(x)+1)(p−1)+1 ∆p u (x + yd(x)/2) d(x)(α(x)+1)(p−1)+1 b (x + yd(x)/2) (u (x + yd(x)/2))q(x+yd(x)/2) A1 d(x)
α(x)(q(x)−q(x+yd(x)/2))
d (x + yd(x)/2) d(x)
µ
v q(x+yd(x)/2)
¯ , there exists a constant C such that in B. Since q is η-Hölder in Ω |q(x) − q (x + yd(x)/2) | ≤ Cd(x)η ,
(3.1)
and hence
∆p v ≥ C v q(x+yd(x)/2) in B for some positive constant C (we are using throughout the paper the letter C to denote constants, that may change from one line to another but are independent of the relevant quantities). That is, v is a subsolution to the equation ∆p v = C v q(x+yd(x)/2) in B. Now we will construct a supersolution to the same equation which blows up on the boundary of B.
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Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
Let φ be the solution to −∆p φ = 1 in B with φ = 0 on ∂ B. For a large positive A0 and some β > 0 to be chosen, we define v¯ = A0 φ −β . Then v¯ will be a supersolution to ∆p v = C v q(x+yd(x)/2) in B provided that q(x+yd(x)/2)−p+1
β p−1 (β + 1)(p − 1)|∇φ|p + β p−1 φ ≤ CA0
φ β(p−1−q(x+yd(x)/2))+p
for all y ∈ B. This inequality can be obtained choosing β large in order to have
β(p − 1 − q(x + yd(x)/2)) + p < 0, and then A0 large enough. This is possible since q(x + yd(x)/2) > p − 1. By comparison, we arrive at v ≤ v¯ in B, and setting y = 0 we obtain u(x) ≤ A0 φ(0)−β d(x)−α(x) for x ∈ V . This shows the upper inequality in (1.3). To prove the lower inequality we take a point x ∈ V ′ and denote by x¯ the closest point to x on ∂ Ω . Modulus an extra reduction of V ′ if necessary it can be assumed that d(¯x + d(x)ν(¯x)) = d(x) for every x ∈ V ′ where ν stands for the outward unit normal and d(x) designates the distance from x to ∂ Ω . Denoting by A the annulus
A = {y ∈ RN : 1 < |y| < 2 + τ }, where τ > 0, we introduce Ax = x¯ + d(x)ν(¯x) + d(x)A and Qx = Ax ∩ Ω (observe that x ∈ Qx , while the annulus Ax is tangent to ∂ Ω at x¯ ). We remark that the outer radius can be any fixed number greater than 2. We can assume, by diminishing the radius of V , that Qx ⊂ V ′ for every x ∈ V . Now define the normalized function
w(y) = d(x)α(x) u(¯x + d(x)ν(¯x) + d(x)y), for y ∈ Qx , where Qx = A ∩ {y ∈ RN : x¯ + d(x)ν(¯x) + d(x)y ∈ Ω }. Then w satisfies ∆p w = d(x)α(x)(p−1)+p b(¯x + d(x)ν(¯x) + d(x)y)uq(¯x+d(x)ν(¯x)+d(x)y) in Qx . Thanks to the Hölder condition verified by q(x) it follows as before that
∆p w ≤ C w q(¯x+d(x)ν(¯x)+d(x)y) in Qx , for a certain positive constant C . On the other hand, it can be seen as before that the problem
∆p z = Cz q(¯x+d(x)ν(¯x)+d(x)y) z=1 z = 0
in A, on |y| = 1, on |y| = 2 + τ ,
has a unique positive solution z. In fact, 0 and 1 are sub- and supersolutions to this problem respectively. Since w ≥ z on
∂ Qx , it follows by comparison that w ≥ z in Qx . Setting y = −2ν(¯x), we arrive at u(x) ≥ z (−2ν(¯x))d(x)−α(x) ,
and the proof of (1.3) concludes by noticing that since z is bounded from below in |y| = 2 we obtain z (−2ν(¯x)) ≥ C > 0, where C is independent of x. In order to get the rigorous blow-up rate at the point x0 ∈ ∂ Ω , we follow the localized argument used in [8,5]. Firstly, the results of the radially symmetric case are given. Lemma 3.1. Consider the singular problem:
N − 1 ′ p−2 ′ ′ p−2 ′ ′ |v | v = a(r )(R − r )µ v q , (|v | v ) + r v ≥ 0, v ′ (0) = 0, lim v(r ) = ∞,
r ∈ (0, R), r ∈ (0, R),
(3.2)
r ↗R
where R > 0, µ ≥ 0, q > p − 1 are three constants, and a ∈ C ([0, R], (0, ∞)). Then for each ε > 0, (3.2) possesses a positive solution vε such that 1 − ε ≤ lim inf r ↗R
vε (r ) M (R − r )−α
≤ lim sup r ↗R
vε ( r ) M (R − r )−α
≤ 1 + ε,
(3.3)
where α and M are defined in (1.6) with κ = a(R). The proof is similar to that of [5, Theorem 4.2]. Here we omit the details. Arguing as the proof of Lemma 3.1, or adapting the latter part of [2, Theorem 1.1], we can get the corresponding result in each of the annuli Aρ,R (x0 ) := {x ∈ RN : 0 < ρ < |x − x0 | < R}.
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
135
Lemma 3.2. Consider the singular problem:
∆p u = a(r )[dist(x, ∂ Aρ,R (x0 ))]µ uq , u = ∞,
x ∈ Aρ,R (x0 ), x ∈ ∂ Aρ,R (x0 ),
(3.4)
where 0 < ρ < R, µ ≥ 0, q > p − 1 and a ∈ C ([ρ, R], (0, ∞)) is the reflection around r = r0 :=
ρ+R 2
of some function
a ∈ C ([r0 , R], (0, ∞)). Then for each ε > 0, the problem (3.4) possesses a positive solution vε such that 1 − ε ≤ lim inf δ(x)↘0
vε (r ) M [δ(x)]−α
≤ lim sup δ(x)↘0
vε (r ) M [δ(x)]−α
≤ 1 + ε,
(3.5)
where α and M are defined in (1.6) with κ = a(R) and
δ(x) := dist(x, ∂ Aρ,R (x0 )) =
R − |x − x0 |, |x − x0 | − ρ,
if r0 ≤ |x − x0 | < R, if ρ ≤ |x − x0 | < r0 .
Proof of Theorem 1.3(ii). Let u be a positive solution of (1.1) and consider x0 ∈ ∂ Ω , κ = κ(x0 ) > 0, µ = µ(x0 ) ≥ 0 and (1.4) holds. Since Ω is of class C 2 , there exist R > 0 and δ0 > 0 such that BR (x0 − (R + δ)nx0 ) ⊂ Ω
for each δ ∈ [0, δ0 ]
(3.6)
and B¯ R (x0 − Rnx0 ) ∩ ∂ Ω = {x0 }. In [5, Figure 4.1], the authors have represented this one-parameter-dependent family of balls. Observe that, in BR (x0 − (R + δ)nx0 ), dist(x, ∂ Ω ) ≥ dist(x, ∂ BR (x0 − (R + δ)nx0 ))
= R − dist(x, x0 − (R + δ)nx0 ) = R − r , where r := |x − [x0 − (R + δ)nx0 ]|. Fix a sufficiently small ϵ0 > 0. Thanks to (1.4), R > 0 can be shortened, if necessary, so that, for each δ ∈ [0, δ0 ], b ≥ (κ − ϵ0 )(R − r )µ ,
q(x) ≥ q − ϵ0 > p − 1
in BR (x0 − (R + δ)nx0 ).
Then for each δ ∈ [0, δ0 ], the restriction uδ := u|BR (x0 −(R+δ)nx0 ) provides us with a positive subsolution of
∆p u = (κ − ϵ0 )(R − r )µ uq−ϵ0 , u = ∞,
x ∈ BR (x0 − (R + δ)nx0 ), x ∈ ∂ BR (x0 − (R + δ)nx0 ).
(3.7)
Thus, any positive solution of (3.7) is a supersolution of the equation of (1.1) that u verifies in BR (x0 − (R + δ)nx0 ). So we have uδ := u|BR (x0 −(R+δ)nx0 ) ≤ Φδ ,
(3.8)
where Φδ is a solution to (3.7). Now, for each sufficiently small ε > 0, let Ψε be any positive radially symmetric solution of
∆p u = (κ − ϵ0 )(R − r )µ uq−ϵ0 , u = ∞,
x ∈ BR (x0 − Rnx0 ), x ∈ ∂ BR (x0 − Rnx0 ).
The existence of Ψε is guaranteed by Lemma 3.1 and by Lemma 3.1 again, we have lim sup r ↗R
ψε (r ) ≤ 1 + ε, Nϵ0 (R − r )−α
µ+p , q − ϵ0 − p + 1 r := |x − [x0 − Rnx0 ]|, α=
(3.9)
Nϵ0 =
p−1
(α + 1)α p−1
κ − ϵ0 Ψε (x) := ψ(r ).
q−ϵ 1−p+1 0
,
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Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
Fix one of those ε ’s and for each sufficiently small δ > 0 consider the function Φδ defined by x ∈ BR (x0 − (R + δ)nx0 ).
Φδ (x) := Ψε (x + δ nx0 ),
By construction, for each sufficiently small δ > 0, Φδ provides us with a large positive solution of (3.7) and, hence, (3.8) implies for each x ∈ BR (x0 − (R + δ)nx0 ) and δ ∈ (0, δ0 ].
u(x) ≤ Ψε (x + δ nx0 ),
Passing to the limit as δ ↘ 0 gives in BR (x0 − Rnx0 )
u ≤ Ψε
and hence for each ω ∈ (0, π /2), (3.9) implies u(x)
lim sup
Nϵ0 [d(x)]−α
x→x0 ,x∈Cx0 ,ω
≤ 1 + ε,
(3.10)
since d(x)
lim
x→x0 ,x∈Cx0 ,ω
=
R−r
d(x)
lim
x→x0 ,x∈Cx0 ,ω
dist(x, ∂ BR (x0 − Rnx0 ))
= 1.
As the estimate (3.10) is valid for any sufficiently small ε > 0 and ϵ0 > 0, for proving (1.5) it remains to show that lim inf
x→x0 ,x∈Cx0 ,ω
u(x) Md(x)−α
≥ 1.
(3.11)
Since Ω is of class C 2 , we may assume a neighborhood W of x0 such that W ∈ C 2 , u > 1 in W . Fix a sufficiently small ϵ0 > 0, µ(x) < µ(x0 ) + ϵ0 in W , and moreover, there exist R2 > R1 > 0 and δ0 > 0 such that AR1 ,R2 (x0 + (R1 + δ)nx0 ) W ⊂ δ∈[0,δ0 ]
and
∂ W ∩ ∂ AR1 ,R2 (x0 + (R1 + δ)nx0 ) = {x0 }. Similarly, R2 can be taken arbitrarily large. Thanks to (1.4), there exists a radially symmetric function bˆ : AR1 ,R2 (x0 + (R1 + δ)nx0 ) → (0, ∞)
such that bˆ ≥ b in W
and for each x ∈ AR1 ,R2 (x0 + (R1 + δ)nx0 ), bˆ (x) = c (|x − x0 − Rnx0 |)[dist(x, ∂ AR1 ,R2 (x0 + (R1 + δ)nx0 ))]µ−ϵ0 for some continuous function c : [R1 , R2 ] → (0, ∞) satisfying c (R1 ) = κ + ϵ0 . Moreover, by enlarging R2 , if necessary, we can assume that c is the reflection around the middle point of [R1 , R2 ] of some continuous positive function. Indeed, it suffices assuming that
|x − x0 − Rnx0 | <
R1 + R2 2
for each x ∈ W .
Furthermore, c can be chosen so that max
AR ,R (x0 +(R1 +δ)nx0 ) 1 2
bˆ ≤ max b + 1. ¯ Ω
Now, consider the auxiliary problem
ˆ q(x0 )+ϵ0 , ∆p u = bu u = ∞,
x ∈ AR1 ,R2 (x0 + R1 nx0 ), x ∈ ∂ AR1 ,R2 (x0 + R1 nx0 ).
(3.12)
Thanks to Lemma 3.2, for each sufficiently small ε > 0, (3.12) possesses a radially symmetric positive solution Ψε such that lim inf r ↓R1
ψε (r ) ≥ 1 − ε, Pϵ0 [r − R1 ]−αˆ
(3.13)
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
137
where
αˆ :=
µ+p , q + ϵ0 − p + 1
Ψε (x) = ψε (r ),
Pϵ0 :=
r := |x − [x0 + R1 nx0 ]|,
p−1
κ + ϵ0
(αˆ + 1)αˆ
p−1
q+ϵ 1−p+1 0
.
Fix one of those ε ’s and for each δ ∈ (0, δ0 ] consider the function Φδ defined by
Φδ (x) := Ψε (x − δ nx0 ),
x ∈ AR1 ,R2 (x0 + (R1 + δ)nx0 ).
For each sufficiently small δ > 0, Φδ provides us with a large positive solution of
∆p u = bˆ (· + δ nx0 )uq(x0 )+ϵ0 , u = ∞,
x ∈ AR1 ,R2 (x0 + (R1 + δ)nx0 ), x ∈ ∂ AR1 ,R2 (x0 + (R1 + δ)nx0 ).
(3.14)
Moreover, by construction, the restriction Φδ on {x ∈ W |Φδ (x) ≥ 1} could provide us with a subsolution of (1.1) in {x ∈ W | Φδ (x) ≥ 1} only if we let W be small enough. Thus, thanks to u > 1 in W and Lemma 2.1, for each δ ∈ (0, δ0 ] we have
Ψε (x − δ nx0 ) ≤ u(x)
for each x ∈ W ∩ AR1 ,R2 (x0 + (R1 + δ)nx0 ) and δ ∈ (0, δ0 ].
Thus, passing to the limit as δ ↓ 0 gives
Ψε ≤ u in W ∩ AR1 ,R2 (x0 + R1 nx0 ), and, hence, for each ω ∈ (0, π /2) u(x)
lim inf
Pϵ0 [d(x)]αˆ
x→x0 ,x∈Cx0 ,ω
≥ 1 − ε,
(3.15)
by d(x)
lim
x→x0 ,x∈Cx0 ,ω
r − R1
=
d(x)
lim
x→x0 ,x∈Cx0 ,ω
This concludes the proof of (1.5).
dist(x, ∂ AR1 ,R2 (x0 + Rnx0 ))
= 1.
Proof of Corollary 1.1. It can be referred to [5, p. 76–78], we omit the details. Now we prove Theorem 1.4. The proof is based on that of Theorem 1.3(i), but taking into account that the exponents here may be variable, and the involved constants have to be precisely estimated. The technique used here can be referred to [21]. Proof of Theorem 1.4. Let ε > 0 and choose a neighborhood W of x0 such that q(y) ≥ p − 1 + (Q − ε)d(y)γ and b(y) ≥ (κ − ε)d(y)µ for y ∈ W . For x close to x0 , and 0 < τ < 1, we have d(y) ≥ (1 − τ )d(x) if y ∈ B(x, τ d(x)) and hence q(y) ≥ p − 1 + (Q − ε)(1 − τ )γ d(x)γ ,
b(y) ≥ (κ − ε)(1 − τ )µ d(x)µ
(3.16)
in B(x, τ d(x)), provided B(x, τ d(x)) ⊂ W , which is certainly true if x is close enough to x0 . Denote for simplicity
σ = σε,τ ,x = (Q − ε)(1 − τ )γ d(x)γ .
(3.17)
If x is close enough to x0 we may further assume that u > 1 in B(x, τ d(x)). Hence
∆p u ≥ (κ − ε)(1 − τ )µ d(x)µ uσ +p−1 in B(x, τ d(x)). We now introduce the function
v(y) = (τ d(x))m u(x + τ d(x)y),
y ∈ B = B(0, 1),
where m=
p+µ
σ
.
(3.18)
Then by calculation, we know v(y) satisfies
∆p v(y) ≥ τ −µ (κ − ε)(1 − τ )µ v(y)σ +p−1 in B. On the other hand, we may look for a supersolution to the equation ∆p v(y) = τ −µ (κ −ε)(1 −τ )µ v(y)σ +p−1 of the form
v¯ = Aφ −β ,
138
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
where
β = p/σ ,
(3.19)
A > 0 and φ is the solution to −∆p φ = 1 in B with φ = 0 on ∂ B. Then v¯ is a supersolution provided that
β p−1 Ap−1 (β + 1)(p − 1)φ −(β+1)(p−1)−1 |∇φ|p + β p−1 Ap−1 φ −(β+1)(p−1) ≤ τ −µ (κ − ε)(1 − τ )µ (Aφ −β )σ +p−1 , that is
β p−1 (β + 1)(p − 1)|∇φ|p + β p−1 φ ≤ τ −µ (κ − ε)(1 − τ )µ Aσ . So by (3.17), it is enough to take A = {Cd(x)}
−γ p σ
for some positive, large enough constant C . By comparison,
v(y) ≤ v¯ (y) if y ∈ B. Setting in particular y = 0 we obtain u(x) ≤ (τ d(x))−m (Cd(x))
−γ p σ
φ(0)−β .
(3.20)
Substituting (3.17)–(3.19) into (3.20), we get that lim sup x→ x0
σ log u(x) ≤ p + pγ + µ. − log d(x)
Letting ε → 0 and then τ → 0, we obtain lim sup x→ x0
d(x)γ log u(x)
− log d(x)
≤
p + pγ + µ Q
.
(3.21)
Next we prove the lower estimate. As in the first part of the proof, we may assume a neighborhood W of x0 has been chosen so that q(y) ≤ p − 1 + (Q + ε)d(y)γ ,
b(y) ≤ (κ + ε)d(y)µ
for y ∈ W . For x close to x0 , we consider the sets A, Ax , Qx , Qx introduced in the proof of Theorem 1.3(i). In Qx we have q(y) ≤ p − 1 + (Q + ε)(1 + τ )γ d(x)γ , and then if u > 1 we have
∆p u ≤ (κ + ε)(1 + τ )µ d(x)µ up−1+θ , where we set
θ = (Q + ε)(1 + τ )γ d(x)γ .
(3.22)
Introduce the function
w(y) = d(x)n u(¯x + d(x)ν(¯x) + d(x)y) p+µ Qx , and for y ∈ Qx , where ν stands for the outward unit normal and n = θ . Then ∆p w(y) ≤ (κ + ε)(1 + τ )µ w(y)p−1+θ in it follows by comparison that w ≥ U in Qx , where U is the unique positive solution to ∆p U = (κ + ε)(1 + τ )µ U p−1+θ , y ∈ A, U = ∞, y ∈ |y| = 1, U = 0, y ∈ |y| = 2 + τ . Thus our next aim will be to estimate from below the solution U when θ → 0. Since U is radial, it verifies U = U (r ), where r = |y| and
N − 1 ′ p−2 ′ (|U ′ |p−2 U ′ )′ + |U | U = (κ + ε)(1 + τ )µ U p−1+θ , r U (1) = ∞, U (2 + τ ) = 0. We introduce the change of variables
ρ = B − Br K ,
r ∈ (1, 2 + τ ),
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
−N +p p−1
where K =
139
and denote V (ρ) = U (r ). Then V verifies (|V ′ |p−2 V ′ )′ = (K B)−p (κ + ε)(1 + τ )µ r (N −1)p/(p−1) V p−1+θ . We
choose B such that (K B)−p (κ + ε)(1 + τ )µ = 1. Denote L = B − B(2 + τ )K , then V satisfies
(N −1)p (|V ′ |p−2 V ′ )′ = r p−1 V p−1+θ , V (0) = ∞, V (L) = 0.
r ∈ (0, L),
Notice that V is convex, and hence thanks to the mean value theorem: V (ρ) = −V ′ (ξ )(L − ρ) ≥ −V ′ (L)(L − ρ),
(3.23)
where ξ ∈ (ρ, L). This shows that it is enough to obtain a lower estimate for −V (L). Since V < 0, we get ′
′
(|V ′ |p−2 V ′ )′ V ′ ≥ cV p−1+θ V ′ , (N −1)p p−1 .
where c := (2 + τ )
An integration in (ρ, L) gives
−V ′ (ρ) |V ′ (L)|p +
pc V (p−1)(p+θ)
(ρ)p+θ
1p ≤ 1.
Integrating with respect to ρ in (0, L) and setting t = V (ρ), we obtain ∞
pc
1p
dt ≤ L. (p − 1)(p + θ ) 1 ∞ 1 (p−1)(p+θ) ′ p p+θ We take t = σ and denote I (θ ) = 0 (1 + σ p+θ )− p dσ . Then, it follows from (3.24) that V ( L ) pc
|V ′ (L)|p +
t p+θ
(3.24)
0
− V ( L) ≥ ′
I (θ )
p+θ θ
(p − 1)(p + θ )
L
θ1
pc
.
(3.25)
On the other hand, if we perform in the integral defining I the change of variable 1 + σ p+θ = t −1 , we obtain I (θ ) =
=
1
1 p+θ
1
p+θ
1
0
1
1
(1 − t ) p+θ −1 t 1 p+θ + p −1
B
1 p+θ
,−
1 p+θ
+
1
p
=
1 p+θ
Γ
1 p+θ
Γ
Γ
1 p
1
−
1 p+θ
,
2
where B and Γ stand for Euler Beta and Gamma functions, respectively. Since Γ (z ) ∼ 1/z as z → 0, it follows that I (θ ) ∼ p/θ as θ → 0, and hence I (θ ) ≤ p/(2θ ) for small θ . This implies, thanks to (3.25), that
1 p p+θ (p − 1)(p + θ ) θ θ , −V (L) ≥ 2θ L pc ′
and then (3.23) gives log V (ρ) ≥
p+θ
θ
log
p 2θ L
+
1
θ
log
(p − 1)(p + θ ) pc
+ log(L − ρ).
Going back to the original variables, we arrive at log U (y) ≥
p+θ
θ
log
p 2θ L
+
1
θ
log
(p − 1)(p + θ ) pc
+ H (|y|),
where H is a function which does not depend on θ . Taking into account that w(y) ≥ u(y), and setting y = −2ν(¯x), we get
θ log u(x) ≥ p + pγ + µ. − log d(x) Letting ε → 0 and then τ → 0, we obtain lim inf x→x0
lim inf x→x0
d(x)γ log u(x)
− log d(x)
≥
p + pγ + µ Q
.
Combining with (3.21), we complete the proof.
We now show the uniqueness. The proof is standard. For the reader’s convenience, we complete it.
140
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
Proof of Theorem 1.5. Let u, v be positive solutions to (1.1). Thanks to Corollary 1.1, lim
x→x0
u(x) =1 v(x)
uniformly for every x0 ∈ ∂ Ω . So for small enough ε > 0 there exists δ > 0 such that
(1 − ε)v ≤ u ≤ (1 + ε)v for all x ∈ Ω such that d(x) ≤ δ . Consider the problem
∆p z = b(x)z q(x) , z = u,
x ∈ Ωδ , x ∈ ∂ Ωδ ,
(3.26)
with Ωδ = {x ∈ Ω : dist(x, ∂ Ω ) > δ}. Problem (3.26) has a unique positive solution, which is precisely u. Now it can be checked that (1 − ε)v and (1 + ε)v are a sub and a supersolution respectively to (3.26), since q(x) ≥ p − 1 in Ω . It follows from the uniqueness of u that (1 − ε)v ≤ u ≤ (1 + ε)v in Ωδ . Thus this inequality is valid throughout Ω , and letting ε → 0 we arrive at u = v , which shows the desired result. Acknowledgments The first author prepared the paper during her visit to College of William and Mary of USA, which was sponsored by Jiangsu Scholarship Council of 2013 in China. She wants to thank Professor Junping Shi for his warm reception and helpfulness. Especially, the authors would like to thank all the anonymous referees for their helpful comments and references. References [1] L. Bieberbach, ∆u = eu und die automorphen Funktionen, Math. Ann. 77 (1916) 173–212. [2] S. Cano-Casanova, J. López-Gómez, Blow-up rates of radially symmetric large solutions, J. Math. Anal. Appl. 352 (2009) 166–174. [3] M. Chuaqui, C. Cortázar, M. Elgueta, J. Garca-Melián, Uniqueness and boundary behaviour of large solutions to elliptic problems with singular weights, Commun. Pure Appl. Anal. 3 (2004) 653–662. [4] F. Cîstea, V. Rˇadulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C. R. Acad. Sci. Paris Sé. I Math. 335 (5) (2002) 447–452. [5] M. Delgado, J. López-Gómez, A. Suárez, Singular boundary value problems of a porous media logistic equation. (English summary), Hiroshima Math. J. 34 (1) (2004) 57–80. [6] J. García-Melián, Uniqueness for boundary blow-up problems with continuous weights, Proc. Amer. Math. Soc. 135 (9) (2007) 2785–2793. [7] J.B. Keller, On solutions of ∆u = f (u), Comm. Pure Appl. Math. 10 (1957) 503–510. [8] J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45. [9] J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2) (2006) 385–439. [10] R. Osserman, On the inequality ∆u ≥ f (u), Pacific J. Math. 7 (1957) 1641–1647. [11] T. Ouyang, Z. Xie, The exact boundary blow-up rate of large solutions for semilinear elliptic problems, Nonlinear Anal. 68 (2008) 2791–2800. [12] Z.J. Zhang, Boundary behavior of solutions to some singular elliptic boundary value problems, Nonlinear Anal. TMA 69 (2008) 2293–2302. [13] Y.J. Chen, M.X. Wang, Boundary blow-up solutions for p-Laplacian elliptic equations of Logistic type, Proc. Roy. Soc. Edingburg 142A (2012) 691–714. [14] Y.H. Du, Z.M. Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. D’Analyse Math. 89 (2003) 277–302. [15] Z. Guo, J.R.L. Webb, Structure of boundary blow-up solutions of quasilinear elliptic problems, II: Small and intermediate solutions, J. Differential Equations 211 (2005) 187–217. [16] J. Matero, Quasilinear elliptic equations with boundary blow-up, J. Anal. Math. 69 (1996) 229–246. [17] A. Mohammed, Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary values, J. Math. Anal. Appl. 325 (2007) 480–489. [18] M. Delgado, J. López-Gómez, A. Suárez, Combining linear and nonlinear diffusion, Adv. Nonlinear Stud. 4 (2004) 273–287. [19] J. López-Gómez, A. Suárez, Combining fast, linear and slow diffusion, Topol. Methods Nonlinear Anal. 23 (2004) 275–300. [20] J. López-Gómez, Varying stoichometric exponents I: classical steady states and metasolutions, Adv. Nonlinear Stud. 3 (2003) 327–354. [21] J. García-Melián, J.D. Rossi, J. Sabina de Lis, Large solutions for the Laplacian with a power nonlinearity given by a variable exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 889–902. [22] J. García-Melián, J.D. Rossi, J. Sabina de Lis, Existence, asymptotic behavior and uniqueness for large solutions to ∆u = eq(x)u , Adv. Nonlinear Stud. 9 (2) (2009) 395–424. [23] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with boundary points, Comm. Partial Differential Equations 8 (1983) 773–817. [24] J. García-Melián, Multiplicity of positive solutions to boundary blow-up elliptic problems with sign-changing weights, J. Funct. Anal. 261 (2011) 1775–1798. [25] A. Cañada, P. Drábek, J.L. Gámez, Existence of positive solutions for some problems with nonlinear diffusion, Trans. Amer. Math. Soc. 349 (1997) 4231–4249.
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Large solutions with a power nonlinearity given by a variable exponent for p-Laplacian equations✩ Yujuan Chen ∗ , Yueping Zhu, Ruiya Hao School of Science, Nantong University, Nantong 226007, PR China
article
abstract
info
Article history: Received 19 April 2014 Accepted 4 August 2014 Communicated by Enzo Mitidieri
In this paper we establish the existence, uniqueness and blow-up rate near the boundary of boundary blow-up solutions to p-Laplacian elliptic equations with a power nonlinearity given by a variable exponent −∆p u = b(x)uq(x) , where ∆p u = div(|∇ u|p−2 ∇ u) with p > 1. Here we combine the localization method originated by López-Gómez (2003) with some previous radially symmetric results to get the asymptotic behavior near the boundary. © 2014 Elsevier Ltd. All rights reserved.
MSC: 35B50 35J65 35B32 Keywords: p-Laplacian equations A variable exponent Boundary blow-up solutions Existence and uniqueness Blow-up rate
1. Introduction and main results In this paper, we will consider the following boundary blow-up problem for p-Laplacian elliptic equations
∆p u = b(x)uq(x) , u = ∞,
x ∈ Ω, x ∈ ∂Ω,
(1.1)
¯ ) will where Ω ⊂ RN (N ≥ 2) is a smooth bounded domain, ∆p u = div(|∇ u|p−2 ∇ u) with p > 1, the exponent q(x) ∈ C η (Ω ¯ ) for some 0 < λ < 1 with b(x) ≥ 0 and b(x) ̸≡ 0 in Ω . Set be a positive continuous function, and b(x) ∈ C λ (Ω Ω0 = int {x ∈ Ω : b(x) = 0}
¯ 0 ⊂ Ω and b(x) > 0 in Ω \ Ω ¯ 0. and suppose, throughout this paper, that Ω 1 ,p 1 By a solution to (1.1) we mean a positive function u ∈ Wloc (Ω ) ∩ C (Ω ) such that (1.1) holds in the distribution sense, and u(x) → ∞ as x ∈ Ω and d(x) → 0, where d(x) = dist(x, ∂ Ω ) represents the distance from x to ∂ Ω for x ∈ Ω . Such a solution of (1.1) is called a boundary blow-up (or large) solution. When q is a constant, the problem (1.1) has been studied by many authors. We quote the papers [1–12] and the references therein for semilinear problems and, [13–17] etc. for problems with the p-Laplacian. However, the main precursor of the
✩ This work was supported by PRC Grants NSFC 11271209, 11371370 and Jiangsu Education Commission 13KJB110023.
∗
Corresponding author. Tel.: +86 0513 85015885. E-mail address: [email protected] (Y. Chen).
http://dx.doi.org/10.1016/j.na.2014.08.003 0362-546X/© 2014 Elsevier Ltd. All rights reserved.
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
131
differential equation of (1.1) in the available mathematical literature is the porous media equation
∆v m(x) = b(x)v, v = 0,
x ∈ Ω, x ∈ ∂Ω,
with variable m(x), which goes back to [18,19]. But, at the best of our knowledge, the previous work concerning large solutions with nonlinearities given a variable exponent is much less. When large solution is concerned, the pioneering paper ¯ and q > 1 in Ω , q = 1 on ∂ Ω , was considered, and the exis [20] where the problem ∆u = −λu + a(x)uq(x) with a > 0 in Ω istence of a maximal and a minimal positive solution was obtained. Then J. García-Melián et al. [21,22] studied the existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem ∆u = uq(x) and ∆u = eq(x)u respectively. As we know, when q is a constant, the necessary and sufficient condition for the existence of the solution to (1.1) is q > p − 1. In our work, however, the function q(x) may be less than p − 1. In addition to considering the existence and nonexistence of positive solutions to (1.1), we also consider the uniqueness and the blow-up rate of solutions near the boundary of the domain. Even though most of the proofs are adaptations of the corresponding ones in [21], we still need to overcome some difficulties. Firstly, sometimes it may not be possible to extend results from p = 2 to p ̸= 2, since many nice features inherent to p = 2 are lost or difficult to verify once p ̸= 2. Secondly, we must face the difficulties caused by the nonnegative weighted function b(x) in the nonlinear term. Therefore, we extend the results in [21,20] to some extent. To prove the existence of the solution to (1.1), we use techniques mainly based on comparison, referring to problem (1.1) with q as a constant. To prove asymptotic behavior near the boundary, we combine the localization method originated by [8] with some radially symmetric results. Now we will state our results. We firstly show that positive solutions to (1.1) are possible only if q(x) ≥ p − 1 on ∂ Ω . Theorem 1.1. Assume that there exists x0 ∈ ∂ Ω such that q(x0 ) < p − 1. Then the problem (1.1) has no positive solutions. ¯. Moreover, the same conclusion holds if q(x) ≤ p − 1 in a whole neighborhood of x0 ∈ ∂ Ω relative to Ω Thanks to Theorem 1.1, we always need q(x) ≥ p − 1 on ∂ Ω in order to have positive solutions. We will make the assumption that q(x) > p − 1 in a neighborhood of ∂ Ω , although q may be p − 1 on ∂ Ω . We also remark that q(x) ≤ p − 1 is permitted at interior points, and we can still get a solution. Theorem 1.2. Assume that q(x) > p−1 in the strip Ωδ = {x ∈ Ω : dist(x, ∂ Ω ) < δ} for some δ > 0. Then problem (1.1) admits at least a positive solution. The following two theorems give the boundary blow-up behavior of positive solutions to (1.1) in the case, where q(x) > p − 1 and q(x) = p − 1 respectively somewhere on ∂ Ω . Subsequently, we denote by n : ∂ Ω → RN , x → n(x) := nx , the outward unit normal vector-field of Ω , and for each ω ∈ (0, π /2), Cx0 ,ω := x ∈ Ω : angle(x − x0 , −nx0 ) ≤ π /2 − ω .
Theorem 1.3. Assume there exists x0 ∈ ∂ Ω such that q(x0 ) > p − 1. Let u be a positive solution to (1.1).
¯ ) and two positive constants A1 and A2 , such that (i) If in the neighborhood of x0 (relative to Ω ), there exists µ(x) ∈ C η (Ω A1 d(x)µ(x) ≤ b(x) ≤ A2 d(x)µ(x)
(1.2)
then there exist a neighborhood V of x0 (relative to Ω ) and positive constants C1 , C2 such that C1 d(x)−α(x) ≤ u(x) ≤ C2 d(x)−α(x)
in V ,
(1.3)
where
µ(x) + p . q(x) − p + 1 (ii) If there exist κ = κ(x0 ) > 0, µ = µ(x0 ) ≥ 0 such that α(x) =
b(x)
= 1, κ d(x)µ then for each ω ∈ (0, π /2), one has that lim
(1.4)
x→x0
lim
x→x0 ,x∈Cx0 ,ω
u(x) Md(x)α(x)
= 1,
(1.5)
where
µ+p α= , q−p+1
M =
1
κ
(p − 1)(α + 1)α
p−1
q−p1+1
.
(1.6)
132
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
Corollary 1.1. Suppose that q(x) > p − 1 on ∂ Ω , and there exist κ ∈ C (∂ Ω , (0, ∞)), µ ∈ C (∂ Ω , (0, ∞)) such that lim
x→x0
b(x) = 1 uniformly in x0 ∈ ∂ Ω . κ(x0 )d(x)µ(x0 )
(1.7)
Then any positive solution u of (1.1) satisfies lim
x→x0
u( x ) M (x0 )d(x)α(x0 )
uniformly in x0 ∈ ∂ Ω ,
= 1,
where
µ(x0 ) + p , α(x0 ) = q(x0 ) − p + 1
M (x0 ) =
1
κ(x0 )
(p − 1)(α(x0 ) + 1)α(x0 )
Therefore, for any pair u1 , u2 of positive solutions of (1.1), limx→x0
p−1
u1 (x) u2 (x)
1 q(x0 )−p+1
.
(1.8)
= 1 uniformly in x0 ∈ ∂ Ω .
Theorem 1.4. Let x0 ∈ ∂ Ω be a point with q(x0 ) = p − 1. If there exist constants µ ≥ 0, κ, γ , Q > 0 such that lim
x→x0
b(x) = 1, κ d(x)µ
lim
q(x) − p + 1 d(x)γ
x→x0
= Q,
(1.9)
then for every positive solution u of (1.1): lim
d(x)γ log u(x)
x→x0
− log d(x)
=
p + pγ + µ Q
.
(1.10)
It is natural to ask under which conditions the solution provided by Theorem 1.2 is unique. It turns out that q > p − 1 on ∂ Ω is sufficient as long as q ≥ p − 1 in the whole Ω . Theorem 1.5. Under the conditions of Corollary 1.1 and q(x) ≥ p − 1 in Ω , the problem (1.1) admits a unique positive solution. The paper is organized as follows: in Section 2 we prove Theorems 1.1 and 1.2. Section 3 will be dedicated to proving the boundary estimates and uniqueness, which are stated in Theorems 1.3–1.5. 2. Existence and nonexistence In this section, we deal with the existence and nonexistence of positive solutions to problem (1.1). First we cite a comparison result for a class of quasilinear equations. Lemma 2.1 (See [23] for a Proof). Let G : Q × R → R be non-increasing in the second variable and continuous. Let u, v ∈ W 1,p (Q) satisfy the respective inequalities
|∇ u|p−2 ∇ u · ∇ϕ ≤ Q
G(x, u)ϕ, Q
and
|∇v|p−2 ∇v · ∇ϕ ≥ Q
G(x, v)ϕ, Q
1,p
for all non-negative ϕ ∈ W0 (Q). Then the inequality u ≤ v on ∂ Q implies u ≤ v in Q. We now show that there are no solutions if q(x) < p − 1 somewhere on ∂ Ω (alternatively, q(x) ≤ p − 1 in a neighborhood of a boundary point). Throughout the paper, we denote by B(x, r ) the ball of center x and radius r.
¯. Proof of Theorem 1.1 (See the Proof of Theorem 1 in [21] or Theorem 1 in [24]). Let r > 0 such that q(x) < p−1 in B(x0 , 3r )∩Ω Now choose a smooth subdomain D of B(x0 , 3r ) ∩ Ω such that ∂ D ∩ ∂ Ω contains B(x0 , 2r ) ∩ ∂ Ω . Let ψ be a smooth function supported on ∂ D which verifies 0 ≤ ψ ≤ 1, ψ = 1 on B(x0 , r ) ∩ ∂ Ω and ψ = 0 on ∂ D \ (B(x0 , 2r ) ∩ ∂ Ω ). Since u = 0 is a subsolution and u = n is a supersolution, the monotone method in [25] then implies the problem ∆p z = b(x)z q(x) , z = nψ,
x ∈ D, x ∈ ∂D
has a positive solution zn for every positive integer n and since q(x) > 0, by Lemma 2.1, it is unique. Moreover, if (1.1) has a positive solution, it follows by comparison that u ≥ zn
in D,
since u ≥ nψ on ∂ D for every n.
(2.1)
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
133
On the other hand, we have zn = nwn , where wn solves
∆p w = nq(x)−p+1 b(x)w q(x) , z = ψ,
x ∈ D, x ∈ ∂ D.
Now 0 ≤ wn ≤ 1 and q(x) < p − 1, so it is standard to conclude (for a subsequence if necessary) that wn → w0 as n → +∞, where w0 is the solution of the following problem
∆p w0 = 0, w = ψ,
x ∈ D, x ∈ ∂ D.
Since w0 > 0 in D, we obtain that zn → +∞ uniformly in compact subsets of D ∪ (B(x0 , r ) ∩ ∂ Ω ). But then (2.1) implies u = +∞ in D ∪ (B(x0 , r ) ∩ ∂ Ω ), which is not possible. Hence no positive solution to (1.1) exists. Finally, observe that the previous argument continues to be valid – with only minor changes – if q(x) ≤ p − 1 in a neighborhood of a point x0 ∈ ∂ Ω . Thus the proof is concluded. Now we prove the existence. The approach is the standard one: we construct solutions with finite datum on ∂ Ω and then show that they are locally uniformly bounded. Proof of Theorem 1.2. Let n be a positive integer. By a similar proof as in Theorem 1.1, the problem
∆p u = b(x)uq(x) , u = n,
x ∈ Ω, x ∈ ∂Ω
(2.2)
has a unique positive solution which is denoted by un . Thanks to un+1 is a supersolution to (2.2) and by uniqueness un+1 ≥ un . Let us prove next that un is bounded in compact subsets of Ω . Taking δ small, we can assume un > 1 in a strip Ωδ = {x ∈ Ω : dist(x, ∂ Ω ) < δ} for all n. Fix ε with 0 < ε < δ and a point x0 such that d(x0 ) = ε/2. Since q(x) > p − 1 in Ωδ , we q have that q(x) ≥ q0 > p − 1 in B(x0 , ε/4) and thus ∆un ≥ b(x)un0 in B(x0 , ε/4). Hence in B(x0 , ε/4), un ≤ U, the positive solution to
∆p U = b(x)U q0 , u = +∞,
x ∈ B(x0 , ε/4), x ∈ ∂ B(x0 , ε/4).
(2.3)
In fact, the existence of the solution to problem (2.3) can be found in [13, Theorem 1.2]. This shows that un is uniformly bounded in B(x0 , ε/8). A compactness argument proves that un is uniformly bounded in the set {x ∈ Ω : d(x) = ε/2}, and since ∆p un ≥ 0 in Ω , we obtain uniform bounds in the whole {x ∈ Ω : d(x) > ε/2}. Since ε was arbitrarily small, the sequence {un } is locally uniformly bounded in Ω . Finally, standard regularity and compactness arguments show that limn→∞ un (x) = u∞ (x) exists and is a positive blow-up solution of (1.1). This finishes the proof. 3. Blow-up rates and uniqueness Firstly, we give a rough estimate of the solutions to (1.1). The idea of the proof comes from [21]. Proof of Theorem 1.3(i). Choose a neighborhood V ′ of x0 such that q(x) > p − 1 in V ′ , meanwhile, (1.2) holds in V ′ (we can take for instance a ball centered at x0 intersected with Ω ). By diminishing the radius of V ′ , we can select a smaller neighborhood V such that B(x, d(x)/2) ⊂ V ′ for x ∈ V . Take x ∈ V and define the scaled function
v(y) = d(x)α(x) u (x + yd(x)/2) , for y ∈ B := B(0, 1). Applying (1.1)–(1.2), it can be checked that the function v satisfies
∆p v =
= ≥
1 2p 1 2p 1 2p
d(x)(α(x)+1)(p−1)+1 ∆p u (x + yd(x)/2) d(x)(α(x)+1)(p−1)+1 b (x + yd(x)/2) (u (x + yd(x)/2))q(x+yd(x)/2) A1 d(x)
α(x)(q(x)−q(x+yd(x)/2))
d (x + yd(x)/2) d(x)
µ
v q(x+yd(x)/2)
¯ , there exists a constant C such that in B. Since q is η-Hölder in Ω |q(x) − q (x + yd(x)/2) | ≤ Cd(x)η ,
(3.1)
and hence
∆p v ≥ C v q(x+yd(x)/2) in B for some positive constant C (we are using throughout the paper the letter C to denote constants, that may change from one line to another but are independent of the relevant quantities). That is, v is a subsolution to the equation ∆p v = C v q(x+yd(x)/2) in B. Now we will construct a supersolution to the same equation which blows up on the boundary of B.
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Let φ be the solution to −∆p φ = 1 in B with φ = 0 on ∂ B. For a large positive A0 and some β > 0 to be chosen, we define v¯ = A0 φ −β . Then v¯ will be a supersolution to ∆p v = C v q(x+yd(x)/2) in B provided that q(x+yd(x)/2)−p+1
β p−1 (β + 1)(p − 1)|∇φ|p + β p−1 φ ≤ CA0
φ β(p−1−q(x+yd(x)/2))+p
for all y ∈ B. This inequality can be obtained choosing β large in order to have
β(p − 1 − q(x + yd(x)/2)) + p < 0, and then A0 large enough. This is possible since q(x + yd(x)/2) > p − 1. By comparison, we arrive at v ≤ v¯ in B, and setting y = 0 we obtain u(x) ≤ A0 φ(0)−β d(x)−α(x) for x ∈ V . This shows the upper inequality in (1.3). To prove the lower inequality we take a point x ∈ V ′ and denote by x¯ the closest point to x on ∂ Ω . Modulus an extra reduction of V ′ if necessary it can be assumed that d(¯x + d(x)ν(¯x)) = d(x) for every x ∈ V ′ where ν stands for the outward unit normal and d(x) designates the distance from x to ∂ Ω . Denoting by A the annulus
A = {y ∈ RN : 1 < |y| < 2 + τ }, where τ > 0, we introduce Ax = x¯ + d(x)ν(¯x) + d(x)A and Qx = Ax ∩ Ω (observe that x ∈ Qx , while the annulus Ax is tangent to ∂ Ω at x¯ ). We remark that the outer radius can be any fixed number greater than 2. We can assume, by diminishing the radius of V , that Qx ⊂ V ′ for every x ∈ V . Now define the normalized function
w(y) = d(x)α(x) u(¯x + d(x)ν(¯x) + d(x)y), for y ∈ Qx , where Qx = A ∩ {y ∈ RN : x¯ + d(x)ν(¯x) + d(x)y ∈ Ω }. Then w satisfies ∆p w = d(x)α(x)(p−1)+p b(¯x + d(x)ν(¯x) + d(x)y)uq(¯x+d(x)ν(¯x)+d(x)y) in Qx . Thanks to the Hölder condition verified by q(x) it follows as before that
∆p w ≤ C w q(¯x+d(x)ν(¯x)+d(x)y) in Qx , for a certain positive constant C . On the other hand, it can be seen as before that the problem
∆p z = Cz q(¯x+d(x)ν(¯x)+d(x)y) z=1 z = 0
in A, on |y| = 1, on |y| = 2 + τ ,
has a unique positive solution z. In fact, 0 and 1 are sub- and supersolutions to this problem respectively. Since w ≥ z on
∂ Qx , it follows by comparison that w ≥ z in Qx . Setting y = −2ν(¯x), we arrive at u(x) ≥ z (−2ν(¯x))d(x)−α(x) ,
and the proof of (1.3) concludes by noticing that since z is bounded from below in |y| = 2 we obtain z (−2ν(¯x)) ≥ C > 0, where C is independent of x. In order to get the rigorous blow-up rate at the point x0 ∈ ∂ Ω , we follow the localized argument used in [8,5]. Firstly, the results of the radially symmetric case are given. Lemma 3.1. Consider the singular problem:
N − 1 ′ p−2 ′ ′ p−2 ′ ′ |v | v = a(r )(R − r )µ v q , (|v | v ) + r v ≥ 0, v ′ (0) = 0, lim v(r ) = ∞,
r ∈ (0, R), r ∈ (0, R),
(3.2)
r ↗R
where R > 0, µ ≥ 0, q > p − 1 are three constants, and a ∈ C ([0, R], (0, ∞)). Then for each ε > 0, (3.2) possesses a positive solution vε such that 1 − ε ≤ lim inf r ↗R
vε (r ) M (R − r )−α
≤ lim sup r ↗R
vε ( r ) M (R − r )−α
≤ 1 + ε,
(3.3)
where α and M are defined in (1.6) with κ = a(R). The proof is similar to that of [5, Theorem 4.2]. Here we omit the details. Arguing as the proof of Lemma 3.1, or adapting the latter part of [2, Theorem 1.1], we can get the corresponding result in each of the annuli Aρ,R (x0 ) := {x ∈ RN : 0 < ρ < |x − x0 | < R}.
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
135
Lemma 3.2. Consider the singular problem:
∆p u = a(r )[dist(x, ∂ Aρ,R (x0 ))]µ uq , u = ∞,
x ∈ Aρ,R (x0 ), x ∈ ∂ Aρ,R (x0 ),
(3.4)
where 0 < ρ < R, µ ≥ 0, q > p − 1 and a ∈ C ([ρ, R], (0, ∞)) is the reflection around r = r0 :=
ρ+R 2
of some function
a ∈ C ([r0 , R], (0, ∞)). Then for each ε > 0, the problem (3.4) possesses a positive solution vε such that 1 − ε ≤ lim inf δ(x)↘0
vε (r ) M [δ(x)]−α
≤ lim sup δ(x)↘0
vε (r ) M [δ(x)]−α
≤ 1 + ε,
(3.5)
where α and M are defined in (1.6) with κ = a(R) and
δ(x) := dist(x, ∂ Aρ,R (x0 )) =
R − |x − x0 |, |x − x0 | − ρ,
if r0 ≤ |x − x0 | < R, if ρ ≤ |x − x0 | < r0 .
Proof of Theorem 1.3(ii). Let u be a positive solution of (1.1) and consider x0 ∈ ∂ Ω , κ = κ(x0 ) > 0, µ = µ(x0 ) ≥ 0 and (1.4) holds. Since Ω is of class C 2 , there exist R > 0 and δ0 > 0 such that BR (x0 − (R + δ)nx0 ) ⊂ Ω
for each δ ∈ [0, δ0 ]
(3.6)
and B¯ R (x0 − Rnx0 ) ∩ ∂ Ω = {x0 }. In [5, Figure 4.1], the authors have represented this one-parameter-dependent family of balls. Observe that, in BR (x0 − (R + δ)nx0 ), dist(x, ∂ Ω ) ≥ dist(x, ∂ BR (x0 − (R + δ)nx0 ))
= R − dist(x, x0 − (R + δ)nx0 ) = R − r , where r := |x − [x0 − (R + δ)nx0 ]|. Fix a sufficiently small ϵ0 > 0. Thanks to (1.4), R > 0 can be shortened, if necessary, so that, for each δ ∈ [0, δ0 ], b ≥ (κ − ϵ0 )(R − r )µ ,
q(x) ≥ q − ϵ0 > p − 1
in BR (x0 − (R + δ)nx0 ).
Then for each δ ∈ [0, δ0 ], the restriction uδ := u|BR (x0 −(R+δ)nx0 ) provides us with a positive subsolution of
∆p u = (κ − ϵ0 )(R − r )µ uq−ϵ0 , u = ∞,
x ∈ BR (x0 − (R + δ)nx0 ), x ∈ ∂ BR (x0 − (R + δ)nx0 ).
(3.7)
Thus, any positive solution of (3.7) is a supersolution of the equation of (1.1) that u verifies in BR (x0 − (R + δ)nx0 ). So we have uδ := u|BR (x0 −(R+δ)nx0 ) ≤ Φδ ,
(3.8)
where Φδ is a solution to (3.7). Now, for each sufficiently small ε > 0, let Ψε be any positive radially symmetric solution of
∆p u = (κ − ϵ0 )(R − r )µ uq−ϵ0 , u = ∞,
x ∈ BR (x0 − Rnx0 ), x ∈ ∂ BR (x0 − Rnx0 ).
The existence of Ψε is guaranteed by Lemma 3.1 and by Lemma 3.1 again, we have lim sup r ↗R
ψε (r ) ≤ 1 + ε, Nϵ0 (R − r )−α
µ+p , q − ϵ0 − p + 1 r := |x − [x0 − Rnx0 ]|, α=
(3.9)
Nϵ0 =
p−1
(α + 1)α p−1
κ − ϵ0 Ψε (x) := ψ(r ).
q−ϵ 1−p+1 0
,
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Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
Fix one of those ε ’s and for each sufficiently small δ > 0 consider the function Φδ defined by x ∈ BR (x0 − (R + δ)nx0 ).
Φδ (x) := Ψε (x + δ nx0 ),
By construction, for each sufficiently small δ > 0, Φδ provides us with a large positive solution of (3.7) and, hence, (3.8) implies for each x ∈ BR (x0 − (R + δ)nx0 ) and δ ∈ (0, δ0 ].
u(x) ≤ Ψε (x + δ nx0 ),
Passing to the limit as δ ↘ 0 gives in BR (x0 − Rnx0 )
u ≤ Ψε
and hence for each ω ∈ (0, π /2), (3.9) implies u(x)
lim sup
Nϵ0 [d(x)]−α
x→x0 ,x∈Cx0 ,ω
≤ 1 + ε,
(3.10)
since d(x)
lim
x→x0 ,x∈Cx0 ,ω
=
R−r
d(x)
lim
x→x0 ,x∈Cx0 ,ω
dist(x, ∂ BR (x0 − Rnx0 ))
= 1.
As the estimate (3.10) is valid for any sufficiently small ε > 0 and ϵ0 > 0, for proving (1.5) it remains to show that lim inf
x→x0 ,x∈Cx0 ,ω
u(x) Md(x)−α
≥ 1.
(3.11)
Since Ω is of class C 2 , we may assume a neighborhood W of x0 such that W ∈ C 2 , u > 1 in W . Fix a sufficiently small ϵ0 > 0, µ(x) < µ(x0 ) + ϵ0 in W , and moreover, there exist R2 > R1 > 0 and δ0 > 0 such that AR1 ,R2 (x0 + (R1 + δ)nx0 ) W ⊂ δ∈[0,δ0 ]
and
∂ W ∩ ∂ AR1 ,R2 (x0 + (R1 + δ)nx0 ) = {x0 }. Similarly, R2 can be taken arbitrarily large. Thanks to (1.4), there exists a radially symmetric function bˆ : AR1 ,R2 (x0 + (R1 + δ)nx0 ) → (0, ∞)
such that bˆ ≥ b in W
and for each x ∈ AR1 ,R2 (x0 + (R1 + δ)nx0 ), bˆ (x) = c (|x − x0 − Rnx0 |)[dist(x, ∂ AR1 ,R2 (x0 + (R1 + δ)nx0 ))]µ−ϵ0 for some continuous function c : [R1 , R2 ] → (0, ∞) satisfying c (R1 ) = κ + ϵ0 . Moreover, by enlarging R2 , if necessary, we can assume that c is the reflection around the middle point of [R1 , R2 ] of some continuous positive function. Indeed, it suffices assuming that
|x − x0 − Rnx0 | <
R1 + R2 2
for each x ∈ W .
Furthermore, c can be chosen so that max
AR ,R (x0 +(R1 +δ)nx0 ) 1 2
bˆ ≤ max b + 1. ¯ Ω
Now, consider the auxiliary problem
ˆ q(x0 )+ϵ0 , ∆p u = bu u = ∞,
x ∈ AR1 ,R2 (x0 + R1 nx0 ), x ∈ ∂ AR1 ,R2 (x0 + R1 nx0 ).
(3.12)
Thanks to Lemma 3.2, for each sufficiently small ε > 0, (3.12) possesses a radially symmetric positive solution Ψε such that lim inf r ↓R1
ψε (r ) ≥ 1 − ε, Pϵ0 [r − R1 ]−αˆ
(3.13)
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
137
where
αˆ :=
µ+p , q + ϵ0 − p + 1
Ψε (x) = ψε (r ),
Pϵ0 :=
r := |x − [x0 + R1 nx0 ]|,
p−1
κ + ϵ0
(αˆ + 1)αˆ
p−1
q+ϵ 1−p+1 0
.
Fix one of those ε ’s and for each δ ∈ (0, δ0 ] consider the function Φδ defined by
Φδ (x) := Ψε (x − δ nx0 ),
x ∈ AR1 ,R2 (x0 + (R1 + δ)nx0 ).
For each sufficiently small δ > 0, Φδ provides us with a large positive solution of
∆p u = bˆ (· + δ nx0 )uq(x0 )+ϵ0 , u = ∞,
x ∈ AR1 ,R2 (x0 + (R1 + δ)nx0 ), x ∈ ∂ AR1 ,R2 (x0 + (R1 + δ)nx0 ).
(3.14)
Moreover, by construction, the restriction Φδ on {x ∈ W |Φδ (x) ≥ 1} could provide us with a subsolution of (1.1) in {x ∈ W | Φδ (x) ≥ 1} only if we let W be small enough. Thus, thanks to u > 1 in W and Lemma 2.1, for each δ ∈ (0, δ0 ] we have
Ψε (x − δ nx0 ) ≤ u(x)
for each x ∈ W ∩ AR1 ,R2 (x0 + (R1 + δ)nx0 ) and δ ∈ (0, δ0 ].
Thus, passing to the limit as δ ↓ 0 gives
Ψε ≤ u in W ∩ AR1 ,R2 (x0 + R1 nx0 ), and, hence, for each ω ∈ (0, π /2) u(x)
lim inf
Pϵ0 [d(x)]αˆ
x→x0 ,x∈Cx0 ,ω
≥ 1 − ε,
(3.15)
by d(x)
lim
x→x0 ,x∈Cx0 ,ω
r − R1
=
d(x)
lim
x→x0 ,x∈Cx0 ,ω
This concludes the proof of (1.5).
dist(x, ∂ AR1 ,R2 (x0 + Rnx0 ))
= 1.
Proof of Corollary 1.1. It can be referred to [5, p. 76–78], we omit the details. Now we prove Theorem 1.4. The proof is based on that of Theorem 1.3(i), but taking into account that the exponents here may be variable, and the involved constants have to be precisely estimated. The technique used here can be referred to [21]. Proof of Theorem 1.4. Let ε > 0 and choose a neighborhood W of x0 such that q(y) ≥ p − 1 + (Q − ε)d(y)γ and b(y) ≥ (κ − ε)d(y)µ for y ∈ W . For x close to x0 , and 0 < τ < 1, we have d(y) ≥ (1 − τ )d(x) if y ∈ B(x, τ d(x)) and hence q(y) ≥ p − 1 + (Q − ε)(1 − τ )γ d(x)γ ,
b(y) ≥ (κ − ε)(1 − τ )µ d(x)µ
(3.16)
in B(x, τ d(x)), provided B(x, τ d(x)) ⊂ W , which is certainly true if x is close enough to x0 . Denote for simplicity
σ = σε,τ ,x = (Q − ε)(1 − τ )γ d(x)γ .
(3.17)
If x is close enough to x0 we may further assume that u > 1 in B(x, τ d(x)). Hence
∆p u ≥ (κ − ε)(1 − τ )µ d(x)µ uσ +p−1 in B(x, τ d(x)). We now introduce the function
v(y) = (τ d(x))m u(x + τ d(x)y),
y ∈ B = B(0, 1),
where m=
p+µ
σ
.
(3.18)
Then by calculation, we know v(y) satisfies
∆p v(y) ≥ τ −µ (κ − ε)(1 − τ )µ v(y)σ +p−1 in B. On the other hand, we may look for a supersolution to the equation ∆p v(y) = τ −µ (κ −ε)(1 −τ )µ v(y)σ +p−1 of the form
v¯ = Aφ −β ,
138
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
where
β = p/σ ,
(3.19)
A > 0 and φ is the solution to −∆p φ = 1 in B with φ = 0 on ∂ B. Then v¯ is a supersolution provided that
β p−1 Ap−1 (β + 1)(p − 1)φ −(β+1)(p−1)−1 |∇φ|p + β p−1 Ap−1 φ −(β+1)(p−1) ≤ τ −µ (κ − ε)(1 − τ )µ (Aφ −β )σ +p−1 , that is
β p−1 (β + 1)(p − 1)|∇φ|p + β p−1 φ ≤ τ −µ (κ − ε)(1 − τ )µ Aσ . So by (3.17), it is enough to take A = {Cd(x)}
−γ p σ
for some positive, large enough constant C . By comparison,
v(y) ≤ v¯ (y) if y ∈ B. Setting in particular y = 0 we obtain u(x) ≤ (τ d(x))−m (Cd(x))
−γ p σ
φ(0)−β .
(3.20)
Substituting (3.17)–(3.19) into (3.20), we get that lim sup x→ x0
σ log u(x) ≤ p + pγ + µ. − log d(x)
Letting ε → 0 and then τ → 0, we obtain lim sup x→ x0
d(x)γ log u(x)
− log d(x)
≤
p + pγ + µ Q
.
(3.21)
Next we prove the lower estimate. As in the first part of the proof, we may assume a neighborhood W of x0 has been chosen so that q(y) ≤ p − 1 + (Q + ε)d(y)γ ,
b(y) ≤ (κ + ε)d(y)µ
for y ∈ W . For x close to x0 , we consider the sets A, Ax , Qx , Qx introduced in the proof of Theorem 1.3(i). In Qx we have q(y) ≤ p − 1 + (Q + ε)(1 + τ )γ d(x)γ , and then if u > 1 we have
∆p u ≤ (κ + ε)(1 + τ )µ d(x)µ up−1+θ , where we set
θ = (Q + ε)(1 + τ )γ d(x)γ .
(3.22)
Introduce the function
w(y) = d(x)n u(¯x + d(x)ν(¯x) + d(x)y) p+µ Qx , and for y ∈ Qx , where ν stands for the outward unit normal and n = θ . Then ∆p w(y) ≤ (κ + ε)(1 + τ )µ w(y)p−1+θ in it follows by comparison that w ≥ U in Qx , where U is the unique positive solution to ∆p U = (κ + ε)(1 + τ )µ U p−1+θ , y ∈ A, U = ∞, y ∈ |y| = 1, U = 0, y ∈ |y| = 2 + τ . Thus our next aim will be to estimate from below the solution U when θ → 0. Since U is radial, it verifies U = U (r ), where r = |y| and
N − 1 ′ p−2 ′ (|U ′ |p−2 U ′ )′ + |U | U = (κ + ε)(1 + τ )µ U p−1+θ , r U (1) = ∞, U (2 + τ ) = 0. We introduce the change of variables
ρ = B − Br K ,
r ∈ (1, 2 + τ ),
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
−N +p p−1
where K =
139
and denote V (ρ) = U (r ). Then V verifies (|V ′ |p−2 V ′ )′ = (K B)−p (κ + ε)(1 + τ )µ r (N −1)p/(p−1) V p−1+θ . We
choose B such that (K B)−p (κ + ε)(1 + τ )µ = 1. Denote L = B − B(2 + τ )K , then V satisfies
(N −1)p (|V ′ |p−2 V ′ )′ = r p−1 V p−1+θ , V (0) = ∞, V (L) = 0.
r ∈ (0, L),
Notice that V is convex, and hence thanks to the mean value theorem: V (ρ) = −V ′ (ξ )(L − ρ) ≥ −V ′ (L)(L − ρ),
(3.23)
where ξ ∈ (ρ, L). This shows that it is enough to obtain a lower estimate for −V (L). Since V < 0, we get ′
′
(|V ′ |p−2 V ′ )′ V ′ ≥ cV p−1+θ V ′ , (N −1)p p−1 .
where c := (2 + τ )
An integration in (ρ, L) gives
−V ′ (ρ) |V ′ (L)|p +
pc V (p−1)(p+θ)
(ρ)p+θ
1p ≤ 1.
Integrating with respect to ρ in (0, L) and setting t = V (ρ), we obtain ∞
pc
1p
dt ≤ L. (p − 1)(p + θ ) 1 ∞ 1 (p−1)(p+θ) ′ p p+θ We take t = σ and denote I (θ ) = 0 (1 + σ p+θ )− p dσ . Then, it follows from (3.24) that V ( L ) pc
|V ′ (L)|p +
t p+θ
(3.24)
0
− V ( L) ≥ ′
I (θ )
p+θ θ
(p − 1)(p + θ )
L
θ1
pc
.
(3.25)
On the other hand, if we perform in the integral defining I the change of variable 1 + σ p+θ = t −1 , we obtain I (θ ) =
=
1
1 p+θ
1
p+θ
1
0
1
1
(1 − t ) p+θ −1 t 1 p+θ + p −1
B
1 p+θ
,−
1 p+θ
+
1
p
=
1 p+θ
Γ
1 p+θ
Γ
Γ
1 p
1
−
1 p+θ
,
2
where B and Γ stand for Euler Beta and Gamma functions, respectively. Since Γ (z ) ∼ 1/z as z → 0, it follows that I (θ ) ∼ p/θ as θ → 0, and hence I (θ ) ≤ p/(2θ ) for small θ . This implies, thanks to (3.25), that
1 p p+θ (p − 1)(p + θ ) θ θ , −V (L) ≥ 2θ L pc ′
and then (3.23) gives log V (ρ) ≥
p+θ
θ
log
p 2θ L
+
1
θ
log
(p − 1)(p + θ ) pc
+ log(L − ρ).
Going back to the original variables, we arrive at log U (y) ≥
p+θ
θ
log
p 2θ L
+
1
θ
log
(p − 1)(p + θ ) pc
+ H (|y|),
where H is a function which does not depend on θ . Taking into account that w(y) ≥ u(y), and setting y = −2ν(¯x), we get
θ log u(x) ≥ p + pγ + µ. − log d(x) Letting ε → 0 and then τ → 0, we obtain lim inf x→x0
lim inf x→x0
d(x)γ log u(x)
− log d(x)
≥
p + pγ + µ Q
.
Combining with (3.21), we complete the proof.
We now show the uniqueness. The proof is standard. For the reader’s convenience, we complete it.
140
Y. Chen et al. / Nonlinear Analysis 110 (2014) 130–140
Proof of Theorem 1.5. Let u, v be positive solutions to (1.1). Thanks to Corollary 1.1, lim
x→x0
u(x) =1 v(x)
uniformly for every x0 ∈ ∂ Ω . So for small enough ε > 0 there exists δ > 0 such that
(1 − ε)v ≤ u ≤ (1 + ε)v for all x ∈ Ω such that d(x) ≤ δ . Consider the problem
∆p z = b(x)z q(x) , z = u,
x ∈ Ωδ , x ∈ ∂ Ωδ ,
(3.26)
with Ωδ = {x ∈ Ω : dist(x, ∂ Ω ) > δ}. Problem (3.26) has a unique positive solution, which is precisely u. Now it can be checked that (1 − ε)v and (1 + ε)v are a sub and a supersolution respectively to (3.26), since q(x) ≥ p − 1 in Ω . It follows from the uniqueness of u that (1 − ε)v ≤ u ≤ (1 + ε)v in Ωδ . Thus this inequality is valid throughout Ω , and letting ε → 0 we arrive at u = v , which shows the desired result. Acknowledgments The first author prepared the paper during her visit to College of William and Mary of USA, which was sponsored by Jiangsu Scholarship Council of 2013 in China. She wants to thank Professor Junping Shi for his warm reception and helpfulness. Especially, the authors would like to thank all the anonymous referees for their helpful comments and references. References [1] L. Bieberbach, ∆u = eu und die automorphen Funktionen, Math. Ann. 77 (1916) 173–212. [2] S. Cano-Casanova, J. López-Gómez, Blow-up rates of radially symmetric large solutions, J. Math. Anal. Appl. 352 (2009) 166–174. [3] M. Chuaqui, C. Cortázar, M. Elgueta, J. Garca-Melián, Uniqueness and boundary behaviour of large solutions to elliptic problems with singular weights, Commun. Pure Appl. Anal. 3 (2004) 653–662. [4] F. Cîstea, V. Rˇadulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C. R. Acad. Sci. Paris Sé. I Math. 335 (5) (2002) 447–452. [5] M. Delgado, J. López-Gómez, A. Suárez, Singular boundary value problems of a porous media logistic equation. (English summary), Hiroshima Math. J. 34 (1) (2004) 57–80. [6] J. García-Melián, Uniqueness for boundary blow-up problems with continuous weights, Proc. Amer. Math. Soc. 135 (9) (2007) 2785–2793. [7] J.B. Keller, On solutions of ∆u = f (u), Comm. Pure Appl. Math. 10 (1957) 503–510. [8] J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45. [9] J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2) (2006) 385–439. [10] R. Osserman, On the inequality ∆u ≥ f (u), Pacific J. Math. 7 (1957) 1641–1647. [11] T. Ouyang, Z. Xie, The exact boundary blow-up rate of large solutions for semilinear elliptic problems, Nonlinear Anal. 68 (2008) 2791–2800. [12] Z.J. Zhang, Boundary behavior of solutions to some singular elliptic boundary value problems, Nonlinear Anal. TMA 69 (2008) 2293–2302. [13] Y.J. Chen, M.X. Wang, Boundary blow-up solutions for p-Laplacian elliptic equations of Logistic type, Proc. Roy. Soc. Edingburg 142A (2012) 691–714. [14] Y.H. Du, Z.M. Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. D’Analyse Math. 89 (2003) 277–302. [15] Z. Guo, J.R.L. Webb, Structure of boundary blow-up solutions of quasilinear elliptic problems, II: Small and intermediate solutions, J. Differential Equations 211 (2005) 187–217. [16] J. Matero, Quasilinear elliptic equations with boundary blow-up, J. Anal. Math. 69 (1996) 229–246. [17] A. Mohammed, Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary values, J. Math. Anal. Appl. 325 (2007) 480–489. [18] M. Delgado, J. López-Gómez, A. Suárez, Combining linear and nonlinear diffusion, Adv. Nonlinear Stud. 4 (2004) 273–287. [19] J. López-Gómez, A. Suárez, Combining fast, linear and slow diffusion, Topol. Methods Nonlinear Anal. 23 (2004) 275–300. [20] J. López-Gómez, Varying stoichometric exponents I: classical steady states and metasolutions, Adv. Nonlinear Stud. 3 (2003) 327–354. [21] J. García-Melián, J.D. Rossi, J. Sabina de Lis, Large solutions for the Laplacian with a power nonlinearity given by a variable exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 889–902. [22] J. García-Melián, J.D. Rossi, J. Sabina de Lis, Existence, asymptotic behavior and uniqueness for large solutions to ∆u = eq(x)u , Adv. Nonlinear Stud. 9 (2) (2009) 395–424. [23] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with boundary points, Comm. Partial Differential Equations 8 (1983) 773–817. [24] J. García-Melián, Multiplicity of positive solutions to boundary blow-up elliptic problems with sign-changing weights, J. Funct. Anal. 261 (2011) 1775–1798. [25] A. Cañada, P. Drábek, J.L. Gámez, Existence of positive solutions for some problems with nonlinear diffusion, Trans. Amer. Math. Soc. 349 (1997) 4231–4249.