- Email: [email protected]
The existence of large solutions of semilinear elliptic equations with negative exponents
Nonlinear Analysis 73 (2010) 1739–1746
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
The existence of large solutions of semilinear elliptic equations with negative exponentsI Lei Wei School of Mathematical Science, Xuzhou Normal University, Xuzhou 221116, PR China
article
abstract
info
Article history: Received 21 January 2010 Accepted 4 May 2010
In this work, we consider semilinear elliptic equations with boundary blow-up whose nonlinearities involve a negative exponent. Combining sub- and super-solution arguments, comparison principles and topological degree theory, we establish the existence of large solutions. Furthermore, we show the existence of a maximal large positive solution. © 2010 Elsevier Ltd. All rights reserved.
MSC: 35J55 35B40 Keywords: Boundary blow-up Eigenvalue Topological degree Maximal positive solution
1. Introduction In this paper we study the existence of positive solutions and the maximal positive solution of elliptic equations with boundary blow-up
−1u = a(x)u−m − b(x)up , u = +∞,
x ∈ Ω, x ∈ ∂Ω,
(1.1)
¯ ) is a positive function. Throughout this paper, where Ω ⊂ RN is a bounded and smooth domain, and the function b ∈ C η (Ω ¯ ). we always suppose p > 1, m > 0, a ∈ C η (Ω In (1.1), u = +∞ on ∂ Ω means that u(x) → +∞ as x → ∂ Ω . Generally, the solutions of (1.1) are often said to be large solutions. Problems related to boundary blow-up have been studied for a long time. Motivated by a geometric problem, Bieberbach in [1], for the cases N = 2 and f (u) = eu , proved that the following problem has a unique solution u ∈ C 2 (Ω )
−1u = f (u), u = +∞,
x ∈ Ω, x ∈ ∂Ω.
So far, there is a large amount of literature on elliptic equations related with boundary blow-up, apart from the following mentioned papers, the reader can also refer to [1–15]. In [16], Du introduced systematically
−1u = a(x)u − b(x)up , u = +∞,
x ∈ Ω, x ∈ ∂Ω
I This work was supported by the National Natural Science Foundation of China 10771032,10771212.
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.05.011
1740
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
and gave some brilliant results involving the existence, uniqueness and asymptotic behaviour of positive solutions. In [17], Delgado, López-Gómez and Suárez studied the following problem
−1u = λu1/m − a(x)up/m , x ∈ Ω , u = +∞, x ∈ ∂Ω, where Ω is a bounded and smooth domain and 1 < m < p, a(x) ≥ 0. Simultaneously, in [17] the open set Ω+ := {x ∈ Ω : ¯ + satisfies Ω 0 ⊂ Ω . In [18], Delgado, a(x) > 0} is connected with boundary ∂ Ω+ of class C 2 , and the open set Ω0 := Ω \ Ω
López-Gómez and Suárez studied the existence of positive solutions of the problem
−1u = W (x)uq − a(x)f (u), u = +∞,
x ∈ Ω, x ∈ ∂Ω,
with 0 < q < 1, for a rather general class of functions f (u) (see Theorem 1.1 in [18]). Few papers involving boundary blow-up deal with such nonlinearities with negative exponents. Since nonlinearities with negative exponents are singular at 0, 0 cannot be a sub-solution of the corresponding equations. Therefore, if a(x) is a negative function or sign-changing function, we have difficulty in finding a sub-solution of the auxiliary problem
−1u = a(x)u−m − b(x)up , u = φ,
x ∈ Ω, x ∈ ∂Ω,
where φ is a positive function in C 1+α (∂ Ω ). In order to overcome the difficulty, we will show the existence of positive solutions of the following auxiliary problem by using topological degree theory under some conditions, and show the uniqueness of solutions which are larger than a constant γ
−1u = λu−m − µup , x ∈ Ω , u = l, x ∈ ∂Ω, where −λ, µ are positive constants.
In this paper, we also need the following lemma for operators involving the Laplacian, which goes back to [19,20].
¯ ). Then the Lemma 1. Assume that Ω ⊂ RN is a bounded domain of class C 2+η , 0 < η < 1, and suppose that a(x) ∈ C η (Ω following assertions are equivalent: (i) L = −∆ + a satisfies the maximum principle, (ii) L = −∆ + a satisfies the strong maximum principle, (iii) σ1 [∆ + a, Ω ] > 0, where σ1 [∆ + a, Ω ] denotes the first eigenvalue of L with a homogeneous Dirichlet boundary condition, ¯ ) such that −1φ + a(x)φ ≥ 0 and φ|∂ Ω ≥ 0 and φ is not a (iv) there exists a positive strict super-solution φ ∈ C 2 (Ω ) ∩ C 1 (Ω solution. In Section 2, we will give the main results involving the existence of positive solutions and the maximal positive solution of (1.1). 2. Existence of positive solutions of (1.1) We need to give some necessary results. Denote γ = unique positive solution wφ satisfying wφ > γ
−1u = λu−m − b(x)up , u = φ,
x ∈ Ω, x ∈ ∂Ω,
1/(m+p) m p
. Firstly, we show that the following problem has a
(2.1)
where λ ∈ R, φ ∈ C 2+η (∂ Ω ), φ(x) > 0 on ∂ Ω , m > 0 and p > 1. The following proposition plays an important role in order to prove the existence of positive solutions of (2.1). Proposition 1. There exist two positive constants µ0 and L0 such that for any −λ ≤ µ ≤ µ0 (where λ < 0) and l ≥ L0 , the following problem has a unique positive solution wl satisfying wl > γ
−1u = λu−m − µup , u = l,
x ∈ Ω, x ∈ ∂Ω.
(2.2)
Proof. Due to the strong maximum principle, it holds that any positive solution of (2.2) is between 0 and l. Denote v = l − u. For the existence of positive solutions of (2.2), we only need to investigate the following problem
−1v = µ(l − v)p − λ(l − v)−m , v = 0,
x ∈ Ω, x ∈ ∂Ω.
(2.3)
¯ ) = {u ∈ C (Ω ¯ ) : u|∂ Ω = 0} and Dl,γ = {C0 (Ω ¯ ) : 0 ≤ v < l − γ }. Then, Dl,γ is a bounded and relatively open Denote C0 (Ω ¯ ¯ ). Define a map by subset of K , where K = {u ∈ C0 (Ω ) : u ≥ 0} is the naturally positive cone of C0 (Ω N (t , w) = (−∆)−1 [µ(l − w)p − λt (l − w)−m ].
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
1741
It is clear that N : [0, 1] × Dl,γ → K is a completely continuous operator by comparison principles and the standard elliptic arguments. Let R0 = supx,y∈Ω |x − y|/2, then there exists x0 ∈ Ω such that Ω ⊂ BR0 (x0 ), where BR0 (x0 ) denotes the open ball of radius R0 centered at x0 . Take R > R0 , by the standard sub- and super-solution arguments,
−1u = −µup , u = l,
x ∈ BR (x0 ), x ∈ ∂ BR (x0 )
(2.4)
has a non-negative solution. Due to the monotone property of the right hand side of the equation in (2.4), we see that (2.4) has a unique non-negative solution, denoted by u˜ l . This implies that u˜ l is radially symmetric for otherwise a different solution could be obtained by rotating u˜ l . Therefore, denote zl (r ) = u˜ l (x), r = |x − x0 |, then
zl00 +
(N − 1)
p
zl0 = µzl ,
r 0 zl (0) = 0,
0 < r < R,
(2.5)
zl (R) = l.
It is convenient to rewrite (2.5) in the form
(r N −1 zl0 )0 = µr N −1 zlp . Integrating this identity from 0 to r yields zl0 (r ) = r 1−N
r
Z
p
sN −1 µzl (s)ds ≥ 0. 0
Therefore, zl (r ) is a non-decreasing function and zl (r ) ≤ µ 0
p r 1−N zl
(r )
r
Z
sN −1 ds = µ 0
r
p
N
zl (r ).
Similarly to the proof of Theorem 6.1 in [16], it follows that l
Z
zl (0)
H (τ , zl (0))
−1/2
dτ ≤ R ≤ N
1/2
l
Z
zl (0)
H (τ , zl (0))−1/2 dτ ,
Rτ
where H (τ , s) = 2 s µξ p dξ . From the left hand side of the above inequality, we have zl (0) > 0 since the integral in the left hand side is not integrable if zl (0) = 0. Furthermore, zl is a positive function. By the uniqueness of positive solution of (2.4) and the standard sub- and super-solution arguments, {zl } is non-decreasing in l. It is well known that the following problem has a unique positive solution, denoted by uˆ (see [16])
−1u = −µup , u = ∞,
x ∈ BR (x0 ), x ∈ ∂ BR (x0 ).
(2.6)
We can easily prove zl (|x − x0 |) ≤ uˆ (x), x ∈ BR (x0 ). Denote liml→∞ zl (r ) = z (r ) and uµ (x) = z (r ) as r = |x − x0 |, by the uniqueness of positive solution of (2.6), then uµ = uˆ . Thus, we have
Z
∞
H (τ , uµ (x0 ))−1/2 dτ ≤ R ≤ N 1/2
uµ (x0 )
R ∞ R τ
Denote f (β) = β that
β
f (β1 ) − f (β2 ) =
2ξ p dξ
Z
∞
−1/2
Z
β1
Z
∞
=
β1
Z
Denote µ1 =
Z
∞ uµ (x0 )
µ
< 0. R R ∞ τ 1 R
γ
τ
2 β1
τ
2
2 γ ξ p dξ
β1
Z
∞ uµ (x0 )
H (τ , uµ (x0 ))−1/2 dτ .
(2.7)
dτ , and we claim that f (β) is decreasing in (0, ∞). Indeed, let 0 < β2 < β1 , it is clear
ξ p dξ ξ dξ p
−1/2
dτ
−1/2 −1/2
dτ − dτ −
Z
∞
Z
τ
2 β2
Z
∞
β2
Z
ξ p dξ
τ +β2 −β1
2 β1
−1/2
β2
dτ
ξ dξ p
−1/2
dτ
2 . We can see
H (τ , uµ (x0 ))−1/2 dτ ≤ R H⇒
1 R
Z
∞
Z
τ
2 uµ (x0 )
uµ (x0 )
!−1/2 ξ p dξ
2 dτ ≤ µ.
So, if µ ≤ 21 , then the minimum uµ (x0 ) of the unique positive solution uµ of (2.6) is larger than γ . Denote L0 = max∂ Ω u0 (x), µ µ where u0 is the unique positive solution of (2.6) with µ = 21 . We firstly consider a special case l = L0 . Suppose µ ≤ 21 .
1742
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
Obviously, u0 is a sub-solution of the following problem
−1u = −µup , u = L0 ,
x ∈ Ω, x ∈ ∂Ω.
(2.8)
It is clear that (u0 , L0 ) is a pair ordered sub- and super-solution of (2.8), which implies that there exists a positive solution of (2.8), denoted by WL0 such that u0 ≤ WL0 ≤ L0 in Ω . Obviously, WL0 is a unique positive solution of (2.8), and we have WL0 > γ . Therefore, the following problem has only one solution belonging to DL0 ,γ
−1v = µ(L0 − v)p , v = 0,
x ∈ Ω, x ∈ ∂Ω.
We claim deg(I − N (0, ·), DL0 ,γ , 0) = 1.
Indeed, define an operator by N˜ (s, v) = (−∆)−1 [sµ(L0 − v)p ], then it is clear that N˜ : [0, 1] × DL0 ,γ → K is completely continuous. Assume that v˜ ∈ DL0 ,γ and s˜ ∈ [0, 1] satisfy v˜ = N˜ (˜s, v˜ ). Clearly, WL0 is a sub-solution of the following problem
−1u = −˜sµup , u = L0 ,
x ∈ Ω, x ∈ ∂Ω.
By sub- and super-solution arguments, we have WL0 ≤ L0 − v˜ , which implies v˜ < L0 − γ . So, deg(I − N˜ (s, ·), DL0 ,γ , 0) is well defined. By homotopy invariant property, we have deg(I − N˜ (1, ·), DL0 ,γ , 0) = deg(I − N˜ (0, ·), DL0 ,γ , 0) = 1.
So, we have deg(I − N (0, ·), DL0 ,γ , 0) = 1.
Next, we prove that deg(I − N (t , ·), DL0 ,γ , 0) is well defined. Define a set T ⊂ [0, 1] by T = t ∈ [0, 1] : ∃vt ∈ DL0 ,γ s.t. vt = N (t , vt ) .
For any t ∈ T , let wt = L0 − vt , then wt (x) ≥ γ in Ω and clearly wt is a positive solution of the following problem
−1u = λtu−m − µup , u = L0 ,
x ∈ Ω, x ∈ ∂Ω.
(2.9)
Since 0 ∈ T , T is not empty, and let t0 = supt ∈T t. By the standard elliptic arguments, we have t0 ∈ T . For any t ∈ T , assume u1 and u2 are both positive solutions of (2.9) and u1 , u2 ≥ γ , then we claim u1 = u2 . Indeed, suppose Ω0 = {x ∈ Ω : u1 (x) > u2 (x)} 6= ∅. By Green’s formula, we have
Z Ω0
|∇ u2 |2 −
Z
Z Ω0
∇ u1 ∇ u2 =
Ω0
Z = Ω0
−1u2 u2 +
Z ∂ Ω0
u2
∂ u2 + ∂ν
Z Ω0
1u1 u2 −
m m µu2 (up1 − up2 ) − t λu2 (u− − u− 1 2 )+
Z ∂ Ω0
u2
Z ∂ Ω0
u2
∂ u1 ∂ν
∂ u2 ∂ u1 − u2 ∂ν ∂ν
(2.10)
and
Z Ω0
|∇ u1 |2 −
Z
Z Ω0
∇ u1 ∇ u2 =
Ω0
Z = Ω0
−1u1 u1 +
Z ∂ Ω0
u1
∂ u1 + ∂ν
Z Ω0
1u2 u1 −
m m µu1 (up2 − up1 ) − t λu1 (u− − u− 2 1 )+
Z ∂ Ω0
u1
Z ∂ Ω0
u1
∂ u2 ∂ν
∂ u1 ∂ u2 − u1 ∂ν ∂ν
.
(2.11)
By adding (2.10) to (2.11), we have
Z Ω0
|∇ u1 − ∇ u2 |2 =
Z Ω0
Z = Ω0
(|∇ u1 |2 + |∇ u2 |2 − 2∇ u1 ∇ u2 ) m m [µ(up1 − up2 )(u2 − u1 ) − t λ(u1 − u2 )(u− − u− 2 1 )].
(2.12)
This implies that
µ
Z Ω0
(u1 − u2 )(up1 − up2 ) ≤ −t λ
Z Ω0
m m (u1 − u2 )(u− − u− 2 1 ).
(2.13)
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
Let h(s) = sp + s−m , then h0 (s) = psp−1 − ms−m−1 . It is clear that h0 (s) > 0 as s > p u1
p u2
1743
1/(p+m) m p
= γ . Due to u1 , u2 ≥ γ , it
holds that − > u2 − u1 in Ω0 . By (2.13) and together with −λ ≤ µ, we attain a contradiction. So we have u1 ≤ u2 in Ω . Similarly, we have u2 ≤ u1 in Ω . Therefore, for any t ∈ T , (2.9) has a unique positive solution wt satisfying wt ≥ γ . It is clear that wt0 is a sub-solution of (2.9) as t ∈ T , so (wt0 , L0 ) is a pair ordered sub and super-solution of (2.9). By the uniqueness of positive solution of (2.9) in L0 − DL0 ,γ (where L0 − DL0 ,γ = {L0 − v : v ∈ DL0 ,γ }), we have wt ≥ wt0 . Assume x¯ ∈ Ω such that wt0 (¯x) = minx∈Ω wt0 (x), then we have
wt0 (¯x) =
Z
−m
−m
p t0
G(¯x, y)(λt0 wt0 (y) − µw (y))dy − −m
ZΩ
= Ω
Z ∂Ω
L0
∂G dSy ∂ν
p t0
G(¯x, y)(λt0 wt0 (y) − µw (y))dy + L0 −m
> L0 − c1 µLp0 + c2 λγ −m ,
(2.14) µ1
where G is the corresponding Green function, c1 , c2 are two positive constants. By L0 > γ , there exists µ0 ≤ 2 such that it holds that wt0 (¯x) > γ as −λ ≤ µ ≤ µ0 , which implies L0 − wt = vt ∈ DL0 ,γ for any t ∈ T . So deg(I − N (t , ·), DL0 ,γ , 0) is well defined. By the homotopy invariant property of topological degree, it holds that deg(I − N (1, ·), DL0 ,γ , 0) = deg(I − N (0, ·), DL0 ,γ , 0) = 1.
So, if 0 < −λ ≤ µ ≤ µ0 , then (2.3) with l = L0 has a positive solution belonging to DL0 ,γ , i.e., (2.2) with l = L0 has a positive solution wL0 such that wL0 > γ . If l ≥ L0 , then (wL0 , l) is a pair ordered sub- and super-solution of (2.2). By the standard sub- and super-solution arguments, there exists at least one positive solution wl of (2.2) such that wL0 ≤ wl ≤ l. Clearly, wl (x) > γ in Ω , and similarly we can prove that (2.2) has a unique solution in l − Dl,γ . For the positive function b in (2.1), denote b0 = supx∈Ω b(x) and b1 = infx∈Ω b(x). Proposition 2. Suppose 0 < −λ ≤ b1 ≤ b0 ≤ µ0 , infx∈∂ Ω φ(x) ≥ L0 (L0 , µ0 are given in Proposition 1). Then (2.1) has a unique positive solution wφ satisfying wφ > γ . Proof. Denote l = inf∂ Ω φ(x), and take µ satisfying b0 ≤ µ ≤ µ0 . By Proposition 1, (2.2) has a unique positive solution wl satisfying wl > γ . Clearly, we can choose a large constant L such that (wl , L) is a pair ordered sub- and super-solution of (2.1). So, (2.1) has at least one positive solution wφ such that wl ≤ wφ ≤ L. Suppose that w1 , w2 are both positive solutions of (2.1) and satisfy w1 , w2 > γ . If Ω1 := {x ∈ Ω : w1 (x) > w2 (x)} 6= ∅. Similarly to the proof of Proposition 1, we have
Z Ω1
|∇w1 − ∇w2 |2 =
Z Ω1
Z
(|∇w1 |2 + |∇w2 |2 − 2∇w1 ∇w2 ) p
= Ω1
p
b(x)(w1 − w2 )(w2 − w1 ) − λ(w1 − w2 )(w2−m − w1−m ),
which implies that
Z
p
Ω1 p
p
b(x)(w1 − w2 )(w1 − w2 ) ≤ −λ
Z Ω1
(w1 − w2 )(w2−m − w1−m ).
(2.15)
p
By w1 − w2 > w2−m − w1−m and b(x) ≥ −λ in Ω1 , (2.15) is impossible. Therefore, (2.1) has a unique solution, which is larger than γ . Proposition 3. If λ ≥ 0, then (2.1) has a unique positive solution. Proof. If λ = 0, it is obvious. We assume λ > 0. Since b is a positive function, we can take a sufficiently small positive constant and a sufficiently large positive constant L such that (, L) is a pair ordered sub- and super-solution of (2.1). Thus, there exists a positive solution wφ satisfying ≤ wφ ≤ L. Let w ˜ φ be an arbitrary positive solution of (2.1). Suppose Ω 0 := {x ∈ Ω : w ˜ φ (x) > wφ (x)} 6= ∅. We suppose Ω 0 is regular, otherwise we can make use of perturbation method to complete proof. By Lemma 1, we have
σ1 [−∆ + b(x)w ˜ φp−1 − λw ˜ φ−m−1 , Ω 0 ] > 0. It is not difficult to see that in Ω 0
−1wφ − λw ˜ φ−m−1 wφ + b(x)w ˜ φp−1 wφ = λwφ (wφ−m−1 − w ˜ φ−m−1 ) + b(x)wφ (w ˜ φp−1 − wφp−1 ) > 0. It holds that
−1wφ − λw ˜ φ−m−1 wφ + b(x)w ˜ φp−1 wφ > −1w ˜ φ − λw ˜ φ−m + b(x)w ˜ φp , wφ = w ˜ φ,
x ∈ Ω 0, x ∈ ∂ Ω 0.
1744
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
¯ , and similarly, we have w By Lemma 1, we have wφ ≥ w ˜ φ in Ω 0 , so we obtain a contradiction. Thus, wφ ≥ w ˜ φ in Ω ˜ φ ≥ wφ ¯ . Therefore, w in Ω ˜ φ = wφ , i.e., the uniqueness is proved. Theorem 1. If 0 < −λ ≤ b1 ≤ b0 ≤ µ0 , then the following problem has a positive solution w ˜ such that w ˜ >γ
−1u = λu−m − b(x)up , u = ∞,
x ∈ Ω, x ∈ ∂Ω.
(2.16)
¯ ) be a positive function, the problem (2.16) has a positive solution. If λ ≥ 0, b ∈ C η (Ω Proof. Suppose 0 < −λ ≤ b1 ≤ b0 ≤ µ0 . By Proposition 2, as l ≥ L0 , the following problem has a unique solution w ˜ l such that w ˜l > γ
−1u = λu−m − b(x)up , u = l,
x ∈ Ω, x ∈ ∂Ω.
(2.17)
˜ l2 are solutions of (2.17) with l = l1 , l2 , respectively, and ˜ l1 , w ˜ l2 holds, where w We claim that if l1 > l2 ≥ L0 , then w ˜ l1 ≥ w ˜ l2 , l1 ) is a pair ordered sub- and super-solution of the following problem ˜ l2 > γ . Indeed, it is clear that (w satisfy w ˜ l1 , w
−1u = λu−m − b(x)up , u = l1 ,
x ∈ Ω, x ∈ ∂Ω.
(2.18)
Since (2.18) has a unique positive solution w ˜ l1 satisfying w ˜ l1 > γ , and by the standard sub- and super-solution arguments, we have w ˜ l1 ≥ w ˜ l2 . Denote the unique positive solution of the following problem by ul
−1u = −b1 up , u = l,
x ∈ Ω, x ∈ ∂Ω.
(2.19)
Clearly, {ul } is non-decreasing in l, and without loss of generality, assume ul → u in C 2 (Ω ) as l → ∞, and it is well known that u is the unique positive solution of the following problem (refer to [16])
−1u = −b1 up , u = ∞,
x ∈ Ω, x ∈ ∂Ω.
By (w ˜ l , l) being a pair ordered sub- and super-solution of (2.19), it holds that w ˜ l ≤ ul , furthermore we have γ < w ˜ l ≤ u. By the standard elliptic arguments, {w ˜ l } has a convergent subsequence in C 2 (Ω ), without loss of generality, we assume w ˜l → w ˜ in C 2 (Ω ) as l → ∞. So, w is a positive solution of (2.16) and w ˜ > γ. Suppose λ ≥ 0. Since we can refer to [16] when λ = 0, we only need to show the case λ > 0. By Proposition 3, we denote the unique positive solution of (2.17) by w ˜ l . It is well known that {w ˜ l } is non-decreasing in l by the standard suband super-solution arguments and together with the uniqueness of positive solutions of (2.17). Denote the unique positive solution of the following problem by u¯
−1u = −b1 up /2, u = ∞,
x ∈ Ω, x ∈ ∂Ω.
Denote 0 = infx∈Ω u¯ (x). For any compact set K ⊂ Ω , there exists δ0 > 0 such that for any δ ∈ (0, δ0 ] it holds that K ⊂ Ωδ , where Ωδ = {x ∈ Ω : d(x, ∂ Ω ) > δ}. Without loss of generality, let l be large enough such that l can be a super-solution of (2.17). Furthermore, we have w ˜ l ≤ l since a sufficiently small positive constant can be a sub-solution of (2.17). Take δ ∈ (0, δ0 ] sufficiently small such that u¯ (x) ≥ l for any x ∈ ∂ Ωδ . So, we obtain u¯ (x) ≥ w ˜ l (x) for each x ∈ ∂ Ωδ . Choose M = max{2λ0 −m−p /b1 , 1}, then Mb1 u¯ p /2 ≥ λ/0m . Hence, we have M p b(x)¯up − Mb1 u¯ p /2 ≥ λ/0m , we also have b(x)M p u¯ p − Mb1 u¯ p /2 ≥ λ/(M m u¯ m ). So, it holds that in Ω
−∆(M u¯ ) = −Mb1 u¯ p /2 ≥ λ/(M u¯ )m − b(x)(M u¯ )p . So, M u¯ is a super-solution of the following problem
−1u = λu−m − b(x)up , u = wl ,
x ∈ Ωδ , x ∈ ∂ Ωδ .
(2.20)
We can take small enough 0 ( 0 ≤ 0 ) such that 0 is a sub-solution of (2.20). By Proposition 3, we have w ˜ l ≤ M u¯ in Ωδ . From K ⊂ Ωδ it follows that for any sufficiently large l, we have w ˜ l (x) ≤ M u¯ (x) in K . By the arbitrariness of K and the
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
1745
standard elliptic arguments, {w ˜ l } has a convergent subsequence in C 2 (Ω ), so without loss of generality, assume that {w ˜ l} 2 itself is convergent to w ˜ in C (Ω ). Therefore, w ˜ is a positive solution of (2.16).
¯ ) and maxx∈Ω¯ |a(x)| = a0 6= 0 (for the case a0 = 0, the problem (1.1) has been studied in some Theorem 2. Suppose a ∈ C η (Ω references). Then there exists µ0 > 0 such that if a0 < b1 ≤ b0 < µ0 , the problem (1.1) has at least one positive solution, which is larger than γ . Furthermore, (1.1) has a maximal positive solution. Proof. We consider the following two auxiliary problems
−1u = −a0 u−m − b(x)up , u = φ,
x ∈ Ω, x ∈ ∂Ω
(2.21)
and
−1u = a0 u−m − b(x)up , u = φ,
x ∈ Ω, x ∈ ∂Ω.
(2.22)
By Proposition 2, if infx∈∂ Ω φ(x) ≥ L0 , then (2.21) has a unique positive solution vφ satisfying vφ > γ . By Proposition 3, it holds that (2.22) has a unique positive solution, denoted by ωφ . Choose sufficiently large L such that L ≥ maxx∈Ω vφ (x) and (vφ , L) can be a pair ordered sub- and super-solution of the problem (2.22). By the standard sub- and super-solution arguments and the uniqueness of the positive solution of (2.22), it is well known that vφ ≤ ωφ . Clearly, (vφ , ωφ ) is a pair ordered sub- and super-solution of
−1u = a(x)u−m − b(x)up , u = φ,
x ∈ Ω, x ∈ ∂Ω,
(2.23)
so, (2.23) has at least one positive solution θφ such that ωφ ≥ θφ ≥ vφ > γ . For convenience, the corresponding positive solutions of the problems (2.21), (2.22) and (2.23) with φ = l are denoted by vl , ωl and θl satisfying vl ≤ θl ≤ ωl in Ω , and assume vl > γ as l ≥ L0 . By {ωl } being non-decreasing in l, denote ω = liml→∞ ωl , and by Theorem 1 we have that ω is a positive solution of (2.22) with φ = ∞. So, we have θl ≤ ω in Ω . So, if l ≥ L0 , it holds that γ < vl ≤ θl ≤ ω in Ω . By the standard elliptic arguments, {θl } has a convergent subsequence in C 2 (Ω ), without loss of generality, we assume θl → θ in C 2 (Ω ) as l → ∞. Then, it is clear that θ is a positive solution of (1.1) and satisfies θ > γ . Now, we show the existence of a maximal positive solution of (1.1). Since (vl , ωl ) is a pair ordered sub- and super-solution of the following problem
−1u = a(x)u−m − b(x)up , u = l,
x ∈ Ω, x ∈ ∂Ω.
(2.24)
Then, there exists a maximal positive solution θl∗ of (2.24) in the order interval [vl , ωl ]. For any positive solution θl0 of (2.24), it is clear that θl0 ≤ ωl , so (max{vl , θl0 }, ωl ) is a pair ordered weak sub- and super-solution of (2.24). Thus, (2.24) has a maximal weak solution θ˜l in the order interval [max{vl , θl0 }, ωl ]. By regularity arguments, we θ˜l is a classical solution. Obviously, we
have vl ≤ θ˜l ≤ ωl , furthermore, we have θ˜l ≤ θl∗ . By the maximality of θ˜l , we have θl0 ≤ θl∗ , that is, θl∗ is a maximal positive solution of (2.24). Similarly, for sufficiently small δ > 0 and sufficiently large constant l we can prove that the following problem has a maximal positive solution, denoted by θl∗,δ
−1u = a(x)u−m − b(x)up , u = l,
x ∈ Ωδ , x ∈ ∂ Ωδ .
(2.25)
Denote the unique positive solution of the following problem by ωl,δ
−1u = a0 u−m − b(x)up , u = l,
x ∈ Ωδ , x ∈ ∂ Ωδ .
(2.26)
It is easy to prove θl∗,δ ≤ ωl,δ . It is well known that {ωl,δ } is non-decreasing in l and denote ωδ = liml→∞ ωl,δ . Similarly to the proof of Theorem 1, it follows that ωδ is a positive solution of
−1u = a0 u−m − b(x)up , u = ∞,
x ∈ Ωδ , x ∈ ∂ Ωδ .
(2.27)
So, we have θl∗,δ ≤ ωδ for any sufficiently large l, and from the standard elliptic theory it follows that {θl∗,δ } has a convergent subsequence in C 2 (Ω ), without loss of generality, we assume {θl∗,δ } itself is convergent to θδ∗ in C 2 (Ω ) as l → ∞. Assume that θ is an arbitrary positive solution of (1.1). For any given compact set K ⊂ Ω , there exists δ 0 > 0 such that K ⊂ Ωδ for each δ ∈ (0, δ 0 ]. Take l > 0 large enough such that l ≥ maxx∈∂ Ωδ θ (x), then θ is a sub-solution of the problem (2.25). Since θ can be a sub-solution of (2.26), it holds θ ≤ ωl,δ . Hence, (θ , ωl,δ ) is a pair ordered sub and super-solution of (2.25), furthermore, it holds θ ≤ θl∗,δ in Ωδ . So, we also have θ ≤ θδ∗ in Ωδ . We claim that {θδ∗ } is non-decreasing about δ ∈ (0, δ 0 ] in
1746
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
the compact set K . Indeed, we assume δ 0 ≥ δ1 > δ2 > 0, and we can choose l ≥ maxx∈∂ Ωδ θδ∗2 (x). Take L > l large enough 1 such that (θδ∗2 , L) is a pair ordered sub- and super-solution of the following problem
−1u = a(x)u−m − b(x)up , u = l,
x ∈ Ωδ 1 , x ∈ ∂ Ωδ1 .
Together with the standard sub- and super-solution arguments, we have θδ∗2 ≤ θδ∗1 ,l in Ωδ1 , so, θδ∗2 ≤ θδ∗1 in Ωδ1 . Furthermore, we have θδ∗2 ≤ θδ∗1 in K . By the arbitrariness of K , θ¯ (x) = limδ→0+ θδ∗ (x) is well defined in Ω . By the standard elliptic theory,
θ¯ is a positive solution of (1.1). Due to θ ≤ θδ∗ in Ωδ , it is clear that θ¯ is a maximal positive solution of (1.1). ¯ ) (without additional conditions), then a sufficiently small positive constant Remark 1. If a, b are positive functions in C η (Ω can be a sub-solution of (1.1), then (1.1) has at least one positive solution. Furthermore, (1.1) has a maximal positive solution and a minimal positive solution.
For Remark 1, similarly to the proof of Theorem 2, the existence of a maximal positive solution can proved. Here, we only go to show that (1.1) has a minimal positive solution. Denote a1 = supx∈Ω a(x). Similarly to the proof of Proposition 3, we can see that the problem (2.23) has a unique positive solution, denoted by θφ . The unique positive solution of (2.23) with φ = l is denoted by θl . By the uniqueness of the positive solution, it is not difficult to see that {θl } is non-decreasing in l. Denote the unique positive solution of the following problem by ξl
−1u = a1 u−m − b(x)up , u = l,
x ∈ Ω, x ∈ ∂Ω.
Similarly to the proof of Theorem 1, denote ξ := liml→∞ ξl (x), then ξ is a positive solution of the following problem
−1u = a1 u−m − b(x)up , u = ∞,
x ∈ Ω, x ∈ ∂Ω.
Hence, we have θl ≤ ξ in Ω . By the standard elliptic theory, {θl } has a convergent subsequence in C 2 (Ω ), without loss of generality, we assume that {θl } itself is convergent to θ in C 2 (Ω ) as l → ∞. Now, we show that θ is a minimal positive solution of (1.1). For an arbitrary positive solution ζ of (1.1), in order to prove θ ≤ ζ , it is sufficient to prove that for any compact set K ⊂ Ω it holds θ ≤ ζ in K . Therefore, we only need to prove that for any large enough constant l it holds θl ≤ ζ in K . For the given K , there exists δ > 0 such that K ⊂ Ωδ for each δ ∈ (0, δ]. For any large enough l, there exists δ ∈ (0, δ] such that θl ≤ ζ on ∂ Ωδ . So, ζ is a super-solution of the following problem
−1u = a(x)u−m − b(x)up , u = θl ,
x ∈ Ωδ , x ∈ ∂ Ωδ .
At the same time, sufficiently small can be a sub-solution. So, by the standard sub- and super-solution arguments and the uniqueness of positive solution of the above problem, we have θl ≤ ζ in Ωδ , furthermore, θl ≤ ζ in K . Acknowledgements The author is grateful to the referee for helpful comments. He is grateful to Professor Mingxin Wang and Professor Jiang Zhu for their help and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
L. Bieberbach, 1u = eu und die autonorphen Funktionen, Math. Ann. 77 (1916) 173–212. J.B. Keller, On solutions of 1u = f (u), Comm. Pure Appl. Math. 10 (1957) 715–724. C. Bandle, M. Marcus, ‘Large’ solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour, J. Anal. Math. 58 (1992) 9–24. Y. Du, Asymptotic behavior and uniqueness results for boundary blow-up solutions, Differential Integral Equations 17 (2004) 819–834. J. García-Melián, Nondegeneracy and uniqueness for boundary blow-up elliptic problems, J. Differential Equations 223 (2006) 208–227. J.V. Goncalves, A. Roncalli, Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput. 182 (2006) 13–23. J. López-Gómez, Large solutions, metasolutions, and asymptotic behavior of the regular positive solutions of sublinear parabolic problems, Electron. J. Differ. Equ. Conf. 5 (2000) 135–171. J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45. J. López-Gómez, Existence and metacoexistence states in competing species models, Houston J. Math. 29 (2003) 483–536. J. López-Gómez, Dynamics of parabolic equations: from classical solutions to metasolutions, Differential Integral Equations 16 (2003) 813–828. J. López-Gómez, Metasolutions: Malthus versus verhulst in population dynamics. A dream of Volterra, in: Handb. Differ. Equ., in: Stationary Partial Differential Equations, vol. 2, Elsevier, North-Holland, Amsterdam, 2005, pp. 211–309. J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385–439. H. Li, M. Wang, Existence and uniqueness of positive solutions to the boundary blow-up problem for an elliptic system, J. Differential Equations 234 (2007) 246–266. R. Osserman, On the inequality 1u ≥ f (u), Pacific J. Math. 7 (1957) 1641–1647. Z. Zhang, Existence of large solutions for a semilinear elliptic problem via explosive sub-supersolutions, Electron. J. Differential Equations 2 (2006) 1–8. Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, World Scientific, 2006. M. Delgado, J. López-Gómez, A. Suárez, Characterizing the existence of large solutions for a class of sublinear problems with nonlinear diffusion, Adv. Differential Equations 7 (2002) 1235–1256. M. Delgado, J. López-Gómez, A. Suárez, Singular boundary value problems of a porous media logistic equation, Hiroshima Math. J. 34 (2004) 57–80. J. López-Gómez, The maximum principle and the existence of principal eigenvalue for some linear weighted boundary value problems, J. Differential Equations 127 (1) (1996) 263–294. J. López-Gómez, M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications, Differential Integral Equations 7 (2) (1994) 383–398.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
The existence of large solutions of semilinear elliptic equations with negative exponentsI Lei Wei School of Mathematical Science, Xuzhou Normal University, Xuzhou 221116, PR China
article
abstract
info
Article history: Received 21 January 2010 Accepted 4 May 2010
In this work, we consider semilinear elliptic equations with boundary blow-up whose nonlinearities involve a negative exponent. Combining sub- and super-solution arguments, comparison principles and topological degree theory, we establish the existence of large solutions. Furthermore, we show the existence of a maximal large positive solution. © 2010 Elsevier Ltd. All rights reserved.
MSC: 35J55 35B40 Keywords: Boundary blow-up Eigenvalue Topological degree Maximal positive solution
1. Introduction In this paper we study the existence of positive solutions and the maximal positive solution of elliptic equations with boundary blow-up
−1u = a(x)u−m − b(x)up , u = +∞,
x ∈ Ω, x ∈ ∂Ω,
(1.1)
¯ ) is a positive function. Throughout this paper, where Ω ⊂ RN is a bounded and smooth domain, and the function b ∈ C η (Ω ¯ ). we always suppose p > 1, m > 0, a ∈ C η (Ω In (1.1), u = +∞ on ∂ Ω means that u(x) → +∞ as x → ∂ Ω . Generally, the solutions of (1.1) are often said to be large solutions. Problems related to boundary blow-up have been studied for a long time. Motivated by a geometric problem, Bieberbach in [1], for the cases N = 2 and f (u) = eu , proved that the following problem has a unique solution u ∈ C 2 (Ω )
−1u = f (u), u = +∞,
x ∈ Ω, x ∈ ∂Ω.
So far, there is a large amount of literature on elliptic equations related with boundary blow-up, apart from the following mentioned papers, the reader can also refer to [1–15]. In [16], Du introduced systematically
−1u = a(x)u − b(x)up , u = +∞,
x ∈ Ω, x ∈ ∂Ω
I This work was supported by the National Natural Science Foundation of China 10771032,10771212.
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.05.011
1740
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
and gave some brilliant results involving the existence, uniqueness and asymptotic behaviour of positive solutions. In [17], Delgado, López-Gómez and Suárez studied the following problem
−1u = λu1/m − a(x)up/m , x ∈ Ω , u = +∞, x ∈ ∂Ω, where Ω is a bounded and smooth domain and 1 < m < p, a(x) ≥ 0. Simultaneously, in [17] the open set Ω+ := {x ∈ Ω : ¯ + satisfies Ω 0 ⊂ Ω . In [18], Delgado, a(x) > 0} is connected with boundary ∂ Ω+ of class C 2 , and the open set Ω0 := Ω \ Ω
López-Gómez and Suárez studied the existence of positive solutions of the problem
−1u = W (x)uq − a(x)f (u), u = +∞,
x ∈ Ω, x ∈ ∂Ω,
with 0 < q < 1, for a rather general class of functions f (u) (see Theorem 1.1 in [18]). Few papers involving boundary blow-up deal with such nonlinearities with negative exponents. Since nonlinearities with negative exponents are singular at 0, 0 cannot be a sub-solution of the corresponding equations. Therefore, if a(x) is a negative function or sign-changing function, we have difficulty in finding a sub-solution of the auxiliary problem
−1u = a(x)u−m − b(x)up , u = φ,
x ∈ Ω, x ∈ ∂Ω,
where φ is a positive function in C 1+α (∂ Ω ). In order to overcome the difficulty, we will show the existence of positive solutions of the following auxiliary problem by using topological degree theory under some conditions, and show the uniqueness of solutions which are larger than a constant γ
−1u = λu−m − µup , x ∈ Ω , u = l, x ∈ ∂Ω, where −λ, µ are positive constants.
In this paper, we also need the following lemma for operators involving the Laplacian, which goes back to [19,20].
¯ ). Then the Lemma 1. Assume that Ω ⊂ RN is a bounded domain of class C 2+η , 0 < η < 1, and suppose that a(x) ∈ C η (Ω following assertions are equivalent: (i) L = −∆ + a satisfies the maximum principle, (ii) L = −∆ + a satisfies the strong maximum principle, (iii) σ1 [∆ + a, Ω ] > 0, where σ1 [∆ + a, Ω ] denotes the first eigenvalue of L with a homogeneous Dirichlet boundary condition, ¯ ) such that −1φ + a(x)φ ≥ 0 and φ|∂ Ω ≥ 0 and φ is not a (iv) there exists a positive strict super-solution φ ∈ C 2 (Ω ) ∩ C 1 (Ω solution. In Section 2, we will give the main results involving the existence of positive solutions and the maximal positive solution of (1.1). 2. Existence of positive solutions of (1.1) We need to give some necessary results. Denote γ = unique positive solution wφ satisfying wφ > γ
−1u = λu−m − b(x)up , u = φ,
x ∈ Ω, x ∈ ∂Ω,
1/(m+p) m p
. Firstly, we show that the following problem has a
(2.1)
where λ ∈ R, φ ∈ C 2+η (∂ Ω ), φ(x) > 0 on ∂ Ω , m > 0 and p > 1. The following proposition plays an important role in order to prove the existence of positive solutions of (2.1). Proposition 1. There exist two positive constants µ0 and L0 such that for any −λ ≤ µ ≤ µ0 (where λ < 0) and l ≥ L0 , the following problem has a unique positive solution wl satisfying wl > γ
−1u = λu−m − µup , u = l,
x ∈ Ω, x ∈ ∂Ω.
(2.2)
Proof. Due to the strong maximum principle, it holds that any positive solution of (2.2) is between 0 and l. Denote v = l − u. For the existence of positive solutions of (2.2), we only need to investigate the following problem
−1v = µ(l − v)p − λ(l − v)−m , v = 0,
x ∈ Ω, x ∈ ∂Ω.
(2.3)
¯ ) = {u ∈ C (Ω ¯ ) : u|∂ Ω = 0} and Dl,γ = {C0 (Ω ¯ ) : 0 ≤ v < l − γ }. Then, Dl,γ is a bounded and relatively open Denote C0 (Ω ¯ ¯ ). Define a map by subset of K , where K = {u ∈ C0 (Ω ) : u ≥ 0} is the naturally positive cone of C0 (Ω N (t , w) = (−∆)−1 [µ(l − w)p − λt (l − w)−m ].
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
1741
It is clear that N : [0, 1] × Dl,γ → K is a completely continuous operator by comparison principles and the standard elliptic arguments. Let R0 = supx,y∈Ω |x − y|/2, then there exists x0 ∈ Ω such that Ω ⊂ BR0 (x0 ), where BR0 (x0 ) denotes the open ball of radius R0 centered at x0 . Take R > R0 , by the standard sub- and super-solution arguments,
−1u = −µup , u = l,
x ∈ BR (x0 ), x ∈ ∂ BR (x0 )
(2.4)
has a non-negative solution. Due to the monotone property of the right hand side of the equation in (2.4), we see that (2.4) has a unique non-negative solution, denoted by u˜ l . This implies that u˜ l is radially symmetric for otherwise a different solution could be obtained by rotating u˜ l . Therefore, denote zl (r ) = u˜ l (x), r = |x − x0 |, then
zl00 +
(N − 1)
p
zl0 = µzl ,
r 0 zl (0) = 0,
0 < r < R,
(2.5)
zl (R) = l.
It is convenient to rewrite (2.5) in the form
(r N −1 zl0 )0 = µr N −1 zlp . Integrating this identity from 0 to r yields zl0 (r ) = r 1−N
r
Z
p
sN −1 µzl (s)ds ≥ 0. 0
Therefore, zl (r ) is a non-decreasing function and zl (r ) ≤ µ 0
p r 1−N zl
(r )
r
Z
sN −1 ds = µ 0
r
p
N
zl (r ).
Similarly to the proof of Theorem 6.1 in [16], it follows that l
Z
zl (0)
H (τ , zl (0))
−1/2
dτ ≤ R ≤ N
1/2
l
Z
zl (0)
H (τ , zl (0))−1/2 dτ ,
Rτ
where H (τ , s) = 2 s µξ p dξ . From the left hand side of the above inequality, we have zl (0) > 0 since the integral in the left hand side is not integrable if zl (0) = 0. Furthermore, zl is a positive function. By the uniqueness of positive solution of (2.4) and the standard sub- and super-solution arguments, {zl } is non-decreasing in l. It is well known that the following problem has a unique positive solution, denoted by uˆ (see [16])
−1u = −µup , u = ∞,
x ∈ BR (x0 ), x ∈ ∂ BR (x0 ).
(2.6)
We can easily prove zl (|x − x0 |) ≤ uˆ (x), x ∈ BR (x0 ). Denote liml→∞ zl (r ) = z (r ) and uµ (x) = z (r ) as r = |x − x0 |, by the uniqueness of positive solution of (2.6), then uµ = uˆ . Thus, we have
Z
∞
H (τ , uµ (x0 ))−1/2 dτ ≤ R ≤ N 1/2
uµ (x0 )
R ∞ R τ
Denote f (β) = β that
β
f (β1 ) − f (β2 ) =
2ξ p dξ
Z
∞
−1/2
Z
β1
Z
∞
=
β1
Z
Denote µ1 =
Z
∞ uµ (x0 )
µ
< 0. R R ∞ τ 1 R
γ
τ
2 β1
τ
2
2 γ ξ p dξ
β1
Z
∞ uµ (x0 )
H (τ , uµ (x0 ))−1/2 dτ .
(2.7)
dτ , and we claim that f (β) is decreasing in (0, ∞). Indeed, let 0 < β2 < β1 , it is clear
ξ p dξ ξ dξ p
−1/2
dτ
−1/2 −1/2
dτ − dτ −
Z
∞
Z
τ
2 β2
Z
∞
β2
Z
ξ p dξ
τ +β2 −β1
2 β1
−1/2
β2
dτ
ξ dξ p
−1/2
dτ
2 . We can see
H (τ , uµ (x0 ))−1/2 dτ ≤ R H⇒
1 R
Z
∞
Z
τ
2 uµ (x0 )
uµ (x0 )
!−1/2 ξ p dξ
2 dτ ≤ µ.
So, if µ ≤ 21 , then the minimum uµ (x0 ) of the unique positive solution uµ of (2.6) is larger than γ . Denote L0 = max∂ Ω u0 (x), µ µ where u0 is the unique positive solution of (2.6) with µ = 21 . We firstly consider a special case l = L0 . Suppose µ ≤ 21 .
1742
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
Obviously, u0 is a sub-solution of the following problem
−1u = −µup , u = L0 ,
x ∈ Ω, x ∈ ∂Ω.
(2.8)
It is clear that (u0 , L0 ) is a pair ordered sub- and super-solution of (2.8), which implies that there exists a positive solution of (2.8), denoted by WL0 such that u0 ≤ WL0 ≤ L0 in Ω . Obviously, WL0 is a unique positive solution of (2.8), and we have WL0 > γ . Therefore, the following problem has only one solution belonging to DL0 ,γ
−1v = µ(L0 − v)p , v = 0,
x ∈ Ω, x ∈ ∂Ω.
We claim deg(I − N (0, ·), DL0 ,γ , 0) = 1.
Indeed, define an operator by N˜ (s, v) = (−∆)−1 [sµ(L0 − v)p ], then it is clear that N˜ : [0, 1] × DL0 ,γ → K is completely continuous. Assume that v˜ ∈ DL0 ,γ and s˜ ∈ [0, 1] satisfy v˜ = N˜ (˜s, v˜ ). Clearly, WL0 is a sub-solution of the following problem
−1u = −˜sµup , u = L0 ,
x ∈ Ω, x ∈ ∂Ω.
By sub- and super-solution arguments, we have WL0 ≤ L0 − v˜ , which implies v˜ < L0 − γ . So, deg(I − N˜ (s, ·), DL0 ,γ , 0) is well defined. By homotopy invariant property, we have deg(I − N˜ (1, ·), DL0 ,γ , 0) = deg(I − N˜ (0, ·), DL0 ,γ , 0) = 1.
So, we have deg(I − N (0, ·), DL0 ,γ , 0) = 1.
Next, we prove that deg(I − N (t , ·), DL0 ,γ , 0) is well defined. Define a set T ⊂ [0, 1] by T = t ∈ [0, 1] : ∃vt ∈ DL0 ,γ s.t. vt = N (t , vt ) .
For any t ∈ T , let wt = L0 − vt , then wt (x) ≥ γ in Ω and clearly wt is a positive solution of the following problem
−1u = λtu−m − µup , u = L0 ,
x ∈ Ω, x ∈ ∂Ω.
(2.9)
Since 0 ∈ T , T is not empty, and let t0 = supt ∈T t. By the standard elliptic arguments, we have t0 ∈ T . For any t ∈ T , assume u1 and u2 are both positive solutions of (2.9) and u1 , u2 ≥ γ , then we claim u1 = u2 . Indeed, suppose Ω0 = {x ∈ Ω : u1 (x) > u2 (x)} 6= ∅. By Green’s formula, we have
Z Ω0
|∇ u2 |2 −
Z
Z Ω0
∇ u1 ∇ u2 =
Ω0
Z = Ω0
−1u2 u2 +
Z ∂ Ω0
u2
∂ u2 + ∂ν
Z Ω0
1u1 u2 −
m m µu2 (up1 − up2 ) − t λu2 (u− − u− 1 2 )+
Z ∂ Ω0
u2
Z ∂ Ω0
u2
∂ u1 ∂ν
∂ u2 ∂ u1 − u2 ∂ν ∂ν
(2.10)
and
Z Ω0
|∇ u1 |2 −
Z
Z Ω0
∇ u1 ∇ u2 =
Ω0
Z = Ω0
−1u1 u1 +
Z ∂ Ω0
u1
∂ u1 + ∂ν
Z Ω0
1u2 u1 −
m m µu1 (up2 − up1 ) − t λu1 (u− − u− 2 1 )+
Z ∂ Ω0
u1
Z ∂ Ω0
u1
∂ u2 ∂ν
∂ u1 ∂ u2 − u1 ∂ν ∂ν
.
(2.11)
By adding (2.10) to (2.11), we have
Z Ω0
|∇ u1 − ∇ u2 |2 =
Z Ω0
Z = Ω0
(|∇ u1 |2 + |∇ u2 |2 − 2∇ u1 ∇ u2 ) m m [µ(up1 − up2 )(u2 − u1 ) − t λ(u1 − u2 )(u− − u− 2 1 )].
(2.12)
This implies that
µ
Z Ω0
(u1 − u2 )(up1 − up2 ) ≤ −t λ
Z Ω0
m m (u1 − u2 )(u− − u− 2 1 ).
(2.13)
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
Let h(s) = sp + s−m , then h0 (s) = psp−1 − ms−m−1 . It is clear that h0 (s) > 0 as s > p u1
p u2
1743
1/(p+m) m p
= γ . Due to u1 , u2 ≥ γ , it
holds that − > u2 − u1 in Ω0 . By (2.13) and together with −λ ≤ µ, we attain a contradiction. So we have u1 ≤ u2 in Ω . Similarly, we have u2 ≤ u1 in Ω . Therefore, for any t ∈ T , (2.9) has a unique positive solution wt satisfying wt ≥ γ . It is clear that wt0 is a sub-solution of (2.9) as t ∈ T , so (wt0 , L0 ) is a pair ordered sub and super-solution of (2.9). By the uniqueness of positive solution of (2.9) in L0 − DL0 ,γ (where L0 − DL0 ,γ = {L0 − v : v ∈ DL0 ,γ }), we have wt ≥ wt0 . Assume x¯ ∈ Ω such that wt0 (¯x) = minx∈Ω wt0 (x), then we have
wt0 (¯x) =
Z
−m
−m
p t0
G(¯x, y)(λt0 wt0 (y) − µw (y))dy − −m
ZΩ
= Ω
Z ∂Ω
L0
∂G dSy ∂ν
p t0
G(¯x, y)(λt0 wt0 (y) − µw (y))dy + L0 −m
> L0 − c1 µLp0 + c2 λγ −m ,
(2.14) µ1
where G is the corresponding Green function, c1 , c2 are two positive constants. By L0 > γ , there exists µ0 ≤ 2 such that it holds that wt0 (¯x) > γ as −λ ≤ µ ≤ µ0 , which implies L0 − wt = vt ∈ DL0 ,γ for any t ∈ T . So deg(I − N (t , ·), DL0 ,γ , 0) is well defined. By the homotopy invariant property of topological degree, it holds that deg(I − N (1, ·), DL0 ,γ , 0) = deg(I − N (0, ·), DL0 ,γ , 0) = 1.
So, if 0 < −λ ≤ µ ≤ µ0 , then (2.3) with l = L0 has a positive solution belonging to DL0 ,γ , i.e., (2.2) with l = L0 has a positive solution wL0 such that wL0 > γ . If l ≥ L0 , then (wL0 , l) is a pair ordered sub- and super-solution of (2.2). By the standard sub- and super-solution arguments, there exists at least one positive solution wl of (2.2) such that wL0 ≤ wl ≤ l. Clearly, wl (x) > γ in Ω , and similarly we can prove that (2.2) has a unique solution in l − Dl,γ . For the positive function b in (2.1), denote b0 = supx∈Ω b(x) and b1 = infx∈Ω b(x). Proposition 2. Suppose 0 < −λ ≤ b1 ≤ b0 ≤ µ0 , infx∈∂ Ω φ(x) ≥ L0 (L0 , µ0 are given in Proposition 1). Then (2.1) has a unique positive solution wφ satisfying wφ > γ . Proof. Denote l = inf∂ Ω φ(x), and take µ satisfying b0 ≤ µ ≤ µ0 . By Proposition 1, (2.2) has a unique positive solution wl satisfying wl > γ . Clearly, we can choose a large constant L such that (wl , L) is a pair ordered sub- and super-solution of (2.1). So, (2.1) has at least one positive solution wφ such that wl ≤ wφ ≤ L. Suppose that w1 , w2 are both positive solutions of (2.1) and satisfy w1 , w2 > γ . If Ω1 := {x ∈ Ω : w1 (x) > w2 (x)} 6= ∅. Similarly to the proof of Proposition 1, we have
Z Ω1
|∇w1 − ∇w2 |2 =
Z Ω1
Z
(|∇w1 |2 + |∇w2 |2 − 2∇w1 ∇w2 ) p
= Ω1
p
b(x)(w1 − w2 )(w2 − w1 ) − λ(w1 − w2 )(w2−m − w1−m ),
which implies that
Z
p
Ω1 p
p
b(x)(w1 − w2 )(w1 − w2 ) ≤ −λ
Z Ω1
(w1 − w2 )(w2−m − w1−m ).
(2.15)
p
By w1 − w2 > w2−m − w1−m and b(x) ≥ −λ in Ω1 , (2.15) is impossible. Therefore, (2.1) has a unique solution, which is larger than γ . Proposition 3. If λ ≥ 0, then (2.1) has a unique positive solution. Proof. If λ = 0, it is obvious. We assume λ > 0. Since b is a positive function, we can take a sufficiently small positive constant and a sufficiently large positive constant L such that (, L) is a pair ordered sub- and super-solution of (2.1). Thus, there exists a positive solution wφ satisfying ≤ wφ ≤ L. Let w ˜ φ be an arbitrary positive solution of (2.1). Suppose Ω 0 := {x ∈ Ω : w ˜ φ (x) > wφ (x)} 6= ∅. We suppose Ω 0 is regular, otherwise we can make use of perturbation method to complete proof. By Lemma 1, we have
σ1 [−∆ + b(x)w ˜ φp−1 − λw ˜ φ−m−1 , Ω 0 ] > 0. It is not difficult to see that in Ω 0
−1wφ − λw ˜ φ−m−1 wφ + b(x)w ˜ φp−1 wφ = λwφ (wφ−m−1 − w ˜ φ−m−1 ) + b(x)wφ (w ˜ φp−1 − wφp−1 ) > 0. It holds that
−1wφ − λw ˜ φ−m−1 wφ + b(x)w ˜ φp−1 wφ > −1w ˜ φ − λw ˜ φ−m + b(x)w ˜ φp , wφ = w ˜ φ,
x ∈ Ω 0, x ∈ ∂ Ω 0.
1744
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
¯ , and similarly, we have w By Lemma 1, we have wφ ≥ w ˜ φ in Ω 0 , so we obtain a contradiction. Thus, wφ ≥ w ˜ φ in Ω ˜ φ ≥ wφ ¯ . Therefore, w in Ω ˜ φ = wφ , i.e., the uniqueness is proved. Theorem 1. If 0 < −λ ≤ b1 ≤ b0 ≤ µ0 , then the following problem has a positive solution w ˜ such that w ˜ >γ
−1u = λu−m − b(x)up , u = ∞,
x ∈ Ω, x ∈ ∂Ω.
(2.16)
¯ ) be a positive function, the problem (2.16) has a positive solution. If λ ≥ 0, b ∈ C η (Ω Proof. Suppose 0 < −λ ≤ b1 ≤ b0 ≤ µ0 . By Proposition 2, as l ≥ L0 , the following problem has a unique solution w ˜ l such that w ˜l > γ
−1u = λu−m − b(x)up , u = l,
x ∈ Ω, x ∈ ∂Ω.
(2.17)
˜ l2 are solutions of (2.17) with l = l1 , l2 , respectively, and ˜ l1 , w ˜ l2 holds, where w We claim that if l1 > l2 ≥ L0 , then w ˜ l1 ≥ w ˜ l2 , l1 ) is a pair ordered sub- and super-solution of the following problem ˜ l2 > γ . Indeed, it is clear that (w satisfy w ˜ l1 , w
−1u = λu−m − b(x)up , u = l1 ,
x ∈ Ω, x ∈ ∂Ω.
(2.18)
Since (2.18) has a unique positive solution w ˜ l1 satisfying w ˜ l1 > γ , and by the standard sub- and super-solution arguments, we have w ˜ l1 ≥ w ˜ l2 . Denote the unique positive solution of the following problem by ul
−1u = −b1 up , u = l,
x ∈ Ω, x ∈ ∂Ω.
(2.19)
Clearly, {ul } is non-decreasing in l, and without loss of generality, assume ul → u in C 2 (Ω ) as l → ∞, and it is well known that u is the unique positive solution of the following problem (refer to [16])
−1u = −b1 up , u = ∞,
x ∈ Ω, x ∈ ∂Ω.
By (w ˜ l , l) being a pair ordered sub- and super-solution of (2.19), it holds that w ˜ l ≤ ul , furthermore we have γ < w ˜ l ≤ u. By the standard elliptic arguments, {w ˜ l } has a convergent subsequence in C 2 (Ω ), without loss of generality, we assume w ˜l → w ˜ in C 2 (Ω ) as l → ∞. So, w is a positive solution of (2.16) and w ˜ > γ. Suppose λ ≥ 0. Since we can refer to [16] when λ = 0, we only need to show the case λ > 0. By Proposition 3, we denote the unique positive solution of (2.17) by w ˜ l . It is well known that {w ˜ l } is non-decreasing in l by the standard suband super-solution arguments and together with the uniqueness of positive solutions of (2.17). Denote the unique positive solution of the following problem by u¯
−1u = −b1 up /2, u = ∞,
x ∈ Ω, x ∈ ∂Ω.
Denote 0 = infx∈Ω u¯ (x). For any compact set K ⊂ Ω , there exists δ0 > 0 such that for any δ ∈ (0, δ0 ] it holds that K ⊂ Ωδ , where Ωδ = {x ∈ Ω : d(x, ∂ Ω ) > δ}. Without loss of generality, let l be large enough such that l can be a super-solution of (2.17). Furthermore, we have w ˜ l ≤ l since a sufficiently small positive constant can be a sub-solution of (2.17). Take δ ∈ (0, δ0 ] sufficiently small such that u¯ (x) ≥ l for any x ∈ ∂ Ωδ . So, we obtain u¯ (x) ≥ w ˜ l (x) for each x ∈ ∂ Ωδ . Choose M = max{2λ0 −m−p /b1 , 1}, then Mb1 u¯ p /2 ≥ λ/0m . Hence, we have M p b(x)¯up − Mb1 u¯ p /2 ≥ λ/0m , we also have b(x)M p u¯ p − Mb1 u¯ p /2 ≥ λ/(M m u¯ m ). So, it holds that in Ω
−∆(M u¯ ) = −Mb1 u¯ p /2 ≥ λ/(M u¯ )m − b(x)(M u¯ )p . So, M u¯ is a super-solution of the following problem
−1u = λu−m − b(x)up , u = wl ,
x ∈ Ωδ , x ∈ ∂ Ωδ .
(2.20)
We can take small enough 0 ( 0 ≤ 0 ) such that 0 is a sub-solution of (2.20). By Proposition 3, we have w ˜ l ≤ M u¯ in Ωδ . From K ⊂ Ωδ it follows that for any sufficiently large l, we have w ˜ l (x) ≤ M u¯ (x) in K . By the arbitrariness of K and the
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
1745
standard elliptic arguments, {w ˜ l } has a convergent subsequence in C 2 (Ω ), so without loss of generality, assume that {w ˜ l} 2 itself is convergent to w ˜ in C (Ω ). Therefore, w ˜ is a positive solution of (2.16).
¯ ) and maxx∈Ω¯ |a(x)| = a0 6= 0 (for the case a0 = 0, the problem (1.1) has been studied in some Theorem 2. Suppose a ∈ C η (Ω references). Then there exists µ0 > 0 such that if a0 < b1 ≤ b0 < µ0 , the problem (1.1) has at least one positive solution, which is larger than γ . Furthermore, (1.1) has a maximal positive solution. Proof. We consider the following two auxiliary problems
−1u = −a0 u−m − b(x)up , u = φ,
x ∈ Ω, x ∈ ∂Ω
(2.21)
and
−1u = a0 u−m − b(x)up , u = φ,
x ∈ Ω, x ∈ ∂Ω.
(2.22)
By Proposition 2, if infx∈∂ Ω φ(x) ≥ L0 , then (2.21) has a unique positive solution vφ satisfying vφ > γ . By Proposition 3, it holds that (2.22) has a unique positive solution, denoted by ωφ . Choose sufficiently large L such that L ≥ maxx∈Ω vφ (x) and (vφ , L) can be a pair ordered sub- and super-solution of the problem (2.22). By the standard sub- and super-solution arguments and the uniqueness of the positive solution of (2.22), it is well known that vφ ≤ ωφ . Clearly, (vφ , ωφ ) is a pair ordered sub- and super-solution of
−1u = a(x)u−m − b(x)up , u = φ,
x ∈ Ω, x ∈ ∂Ω,
(2.23)
so, (2.23) has at least one positive solution θφ such that ωφ ≥ θφ ≥ vφ > γ . For convenience, the corresponding positive solutions of the problems (2.21), (2.22) and (2.23) with φ = l are denoted by vl , ωl and θl satisfying vl ≤ θl ≤ ωl in Ω , and assume vl > γ as l ≥ L0 . By {ωl } being non-decreasing in l, denote ω = liml→∞ ωl , and by Theorem 1 we have that ω is a positive solution of (2.22) with φ = ∞. So, we have θl ≤ ω in Ω . So, if l ≥ L0 , it holds that γ < vl ≤ θl ≤ ω in Ω . By the standard elliptic arguments, {θl } has a convergent subsequence in C 2 (Ω ), without loss of generality, we assume θl → θ in C 2 (Ω ) as l → ∞. Then, it is clear that θ is a positive solution of (1.1) and satisfies θ > γ . Now, we show the existence of a maximal positive solution of (1.1). Since (vl , ωl ) is a pair ordered sub- and super-solution of the following problem
−1u = a(x)u−m − b(x)up , u = l,
x ∈ Ω, x ∈ ∂Ω.
(2.24)
Then, there exists a maximal positive solution θl∗ of (2.24) in the order interval [vl , ωl ]. For any positive solution θl0 of (2.24), it is clear that θl0 ≤ ωl , so (max{vl , θl0 }, ωl ) is a pair ordered weak sub- and super-solution of (2.24). Thus, (2.24) has a maximal weak solution θ˜l in the order interval [max{vl , θl0 }, ωl ]. By regularity arguments, we θ˜l is a classical solution. Obviously, we
have vl ≤ θ˜l ≤ ωl , furthermore, we have θ˜l ≤ θl∗ . By the maximality of θ˜l , we have θl0 ≤ θl∗ , that is, θl∗ is a maximal positive solution of (2.24). Similarly, for sufficiently small δ > 0 and sufficiently large constant l we can prove that the following problem has a maximal positive solution, denoted by θl∗,δ
−1u = a(x)u−m − b(x)up , u = l,
x ∈ Ωδ , x ∈ ∂ Ωδ .
(2.25)
Denote the unique positive solution of the following problem by ωl,δ
−1u = a0 u−m − b(x)up , u = l,
x ∈ Ωδ , x ∈ ∂ Ωδ .
(2.26)
It is easy to prove θl∗,δ ≤ ωl,δ . It is well known that {ωl,δ } is non-decreasing in l and denote ωδ = liml→∞ ωl,δ . Similarly to the proof of Theorem 1, it follows that ωδ is a positive solution of
−1u = a0 u−m − b(x)up , u = ∞,
x ∈ Ωδ , x ∈ ∂ Ωδ .
(2.27)
So, we have θl∗,δ ≤ ωδ for any sufficiently large l, and from the standard elliptic theory it follows that {θl∗,δ } has a convergent subsequence in C 2 (Ω ), without loss of generality, we assume {θl∗,δ } itself is convergent to θδ∗ in C 2 (Ω ) as l → ∞. Assume that θ is an arbitrary positive solution of (1.1). For any given compact set K ⊂ Ω , there exists δ 0 > 0 such that K ⊂ Ωδ for each δ ∈ (0, δ 0 ]. Take l > 0 large enough such that l ≥ maxx∈∂ Ωδ θ (x), then θ is a sub-solution of the problem (2.25). Since θ can be a sub-solution of (2.26), it holds θ ≤ ωl,δ . Hence, (θ , ωl,δ ) is a pair ordered sub and super-solution of (2.25), furthermore, it holds θ ≤ θl∗,δ in Ωδ . So, we also have θ ≤ θδ∗ in Ωδ . We claim that {θδ∗ } is non-decreasing about δ ∈ (0, δ 0 ] in
1746
L. Wei / Nonlinear Analysis 73 (2010) 1739–1746
the compact set K . Indeed, we assume δ 0 ≥ δ1 > δ2 > 0, and we can choose l ≥ maxx∈∂ Ωδ θδ∗2 (x). Take L > l large enough 1 such that (θδ∗2 , L) is a pair ordered sub- and super-solution of the following problem
−1u = a(x)u−m − b(x)up , u = l,
x ∈ Ωδ 1 , x ∈ ∂ Ωδ1 .
Together with the standard sub- and super-solution arguments, we have θδ∗2 ≤ θδ∗1 ,l in Ωδ1 , so, θδ∗2 ≤ θδ∗1 in Ωδ1 . Furthermore, we have θδ∗2 ≤ θδ∗1 in K . By the arbitrariness of K , θ¯ (x) = limδ→0+ θδ∗ (x) is well defined in Ω . By the standard elliptic theory,
θ¯ is a positive solution of (1.1). Due to θ ≤ θδ∗ in Ωδ , it is clear that θ¯ is a maximal positive solution of (1.1). ¯ ) (without additional conditions), then a sufficiently small positive constant Remark 1. If a, b are positive functions in C η (Ω can be a sub-solution of (1.1), then (1.1) has at least one positive solution. Furthermore, (1.1) has a maximal positive solution and a minimal positive solution.
For Remark 1, similarly to the proof of Theorem 2, the existence of a maximal positive solution can proved. Here, we only go to show that (1.1) has a minimal positive solution. Denote a1 = supx∈Ω a(x). Similarly to the proof of Proposition 3, we can see that the problem (2.23) has a unique positive solution, denoted by θφ . The unique positive solution of (2.23) with φ = l is denoted by θl . By the uniqueness of the positive solution, it is not difficult to see that {θl } is non-decreasing in l. Denote the unique positive solution of the following problem by ξl
−1u = a1 u−m − b(x)up , u = l,
x ∈ Ω, x ∈ ∂Ω.
Similarly to the proof of Theorem 1, denote ξ := liml→∞ ξl (x), then ξ is a positive solution of the following problem
−1u = a1 u−m − b(x)up , u = ∞,
x ∈ Ω, x ∈ ∂Ω.
Hence, we have θl ≤ ξ in Ω . By the standard elliptic theory, {θl } has a convergent subsequence in C 2 (Ω ), without loss of generality, we assume that {θl } itself is convergent to θ in C 2 (Ω ) as l → ∞. Now, we show that θ is a minimal positive solution of (1.1). For an arbitrary positive solution ζ of (1.1), in order to prove θ ≤ ζ , it is sufficient to prove that for any compact set K ⊂ Ω it holds θ ≤ ζ in K . Therefore, we only need to prove that for any large enough constant l it holds θl ≤ ζ in K . For the given K , there exists δ > 0 such that K ⊂ Ωδ for each δ ∈ (0, δ]. For any large enough l, there exists δ ∈ (0, δ] such that θl ≤ ζ on ∂ Ωδ . So, ζ is a super-solution of the following problem
−1u = a(x)u−m − b(x)up , u = θl ,
x ∈ Ωδ , x ∈ ∂ Ωδ .
At the same time, sufficiently small can be a sub-solution. So, by the standard sub- and super-solution arguments and the uniqueness of positive solution of the above problem, we have θl ≤ ζ in Ωδ , furthermore, θl ≤ ζ in K . Acknowledgements The author is grateful to the referee for helpful comments. He is grateful to Professor Mingxin Wang and Professor Jiang Zhu for their help and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
L. Bieberbach, 1u = eu und die autonorphen Funktionen, Math. Ann. 77 (1916) 173–212. J.B. Keller, On solutions of 1u = f (u), Comm. Pure Appl. Math. 10 (1957) 715–724. C. Bandle, M. Marcus, ‘Large’ solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour, J. Anal. Math. 58 (1992) 9–24. Y. Du, Asymptotic behavior and uniqueness results for boundary blow-up solutions, Differential Integral Equations 17 (2004) 819–834. J. García-Melián, Nondegeneracy and uniqueness for boundary blow-up elliptic problems, J. Differential Equations 223 (2006) 208–227. J.V. Goncalves, A. Roncalli, Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput. 182 (2006) 13–23. J. López-Gómez, Large solutions, metasolutions, and asymptotic behavior of the regular positive solutions of sublinear parabolic problems, Electron. J. Differ. Equ. Conf. 5 (2000) 135–171. J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45. J. López-Gómez, Existence and metacoexistence states in competing species models, Houston J. Math. 29 (2003) 483–536. J. López-Gómez, Dynamics of parabolic equations: from classical solutions to metasolutions, Differential Integral Equations 16 (2003) 813–828. J. López-Gómez, Metasolutions: Malthus versus verhulst in population dynamics. A dream of Volterra, in: Handb. Differ. Equ., in: Stationary Partial Differential Equations, vol. 2, Elsevier, North-Holland, Amsterdam, 2005, pp. 211–309. J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385–439. H. Li, M. Wang, Existence and uniqueness of positive solutions to the boundary blow-up problem for an elliptic system, J. Differential Equations 234 (2007) 246–266. R. Osserman, On the inequality 1u ≥ f (u), Pacific J. Math. 7 (1957) 1641–1647. Z. Zhang, Existence of large solutions for a semilinear elliptic problem via explosive sub-supersolutions, Electron. J. Differential Equations 2 (2006) 1–8. Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, World Scientific, 2006. M. Delgado, J. López-Gómez, A. Suárez, Characterizing the existence of large solutions for a class of sublinear problems with nonlinear diffusion, Adv. Differential Equations 7 (2002) 1235–1256. M. Delgado, J. López-Gómez, A. Suárez, Singular boundary value problems of a porous media logistic equation, Hiroshima Math. J. 34 (2004) 57–80. J. López-Gómez, The maximum principle and the existence of principal eigenvalue for some linear weighted boundary value problems, J. Differential Equations 127 (1) (1996) 263–294. J. López-Gómez, M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications, Differential Integral Equations 7 (2) (1994) 383–398.