In this section, we will apply the main results to study nonlinear problems which include nonlinear integral equations and nonlinear elliptic boundary value problems for the Lane-Emden-Fowler equations. And then we will obtain new results on the existence and uniqueness of positive solutions for these problems, which are not consequences of the corresponding fixed point theorems in the literature.
3.1 Applications to nonlinear integral equations
A standard approach, in studying the existence of positive solutions of boundary value problems (BVPs for short) for ordinary differential equations, is to rewrite the problem as an equivalent positive-solution problem for a Hammerstein integral equation of the form
(3.1)
in the space , where the nonlinearity f and the kernel G (the Green function of the problem) are both nonnegative, is a parameter. One seeks fixed points of a Hammerstein integral operator in a suitable cone of positive functions.
Set
, the standard cone. It is easy to see that
P is a normal cone of which the normality constant is 1. Then
. Assume that
is continuous with
and there exist
with
, such that
(3.2)
Theorem 3.1
Assume that
and
(H31) is continuous (), is increasing in for fixed and is decreasing in for fixed ;
(H
32)
for ,
there exist (
)
such that
Then,
for any given ,
the integral equation (3.1)
has a unique positive solution in .
Moreover,
for any initial values ,
constructing successively the sequences:
we have , as . Further, (i) if () for , then is strictly increasing in λ, that is, implies ; (ii) if there exists such that () for , then is continuous in λ, that is, () implies ; (iii) if there exists such that () for , then , .
Proof Define two operators
and
by
It is easy to see that u is the solution of (3.1) if and only if . From (H31), we know that is increasing and is decreasing. Further, from (H32), we can prove that A, B satisfy (H11). Next we prove that . Set , . Then .
For any
, from (H
31) and (3.2), we have
Let
,
. Note that
is continuous with
and from (3.2), we get
and in consequence,
. That is,
. Hence, all the conditions of Theorem 2.1 are satisfied. It follows from Theorem 2.1 and Corollary 2.5 that the operator equation
has a unique solution
in
, that is,
. So
is a unique positive solution of the integral equation (
3.1) in
for given
. From Corollary 2.5, we have (i) if
(
) for
, then
is strictly increasing in
λ, that is,
implies
; (ii) if there exists
such that
(
) for
, then
is continuous in
λ, that is,
(
) implies
; (iii) if there exists
such that
(
) for
, then
,
.
Let
,
. Then
,
also satisfy the conditions of Theorem 2.1. By Theorem 2.1, for any initial values
, constructing successively the sequences
we have
,
as
. That is,
as . □
Theorem 3.2 Assume that with satisfies (H31) and
(H33) is continuous, increasing in for fixed , , decreasing in for fixed , ;
(H
34)
for ,
there exist (
)
such that
Then,
for any given ,
the integral equation (3.1)
has a unique positive solution in .
Moreover,
for any initial values ,
constructing successively the sequences:
we have , as . Further, the conclusions (i), (ii), and (iii) in Theorem 3.1 also hold.
Proof Define two operators
and
by
It is easy to see that u is the solution of (3.1) if and only if . From (H31) and (H33), we know that is increasing and is mixed monotone. Further, from (H34), we can prove that A, B satisfy (H21) and (H22). Next we prove that .
For any
, from (H
31), (H
33), and (3.2), we have
Let
,
. Note that
is nonnegative and continuous with
and from (3.2), we get
and in consequence,
. That is,
. Hence, all the conditions of Theorem 2.7 are satisfied. It follows from Theorem 2.7 and Corollary 2.11 that the operator equation
has a unique solution
in
, that is,
. So
is a unique positive solution of the integral equation (
3.1) in
for given
. From Corollary 2.11, we have (i) if
(
) for
, then
is strictly increasing in
λ, that is,
implies
; (ii) if there exists
such that
(
) for
, then
is continuous in
λ, that is,
(
) implies
; (iii) if there exists
such that
(
) for
, then
,
.
Let
,
. Then
,
also satisfy the conditions of Theorem 2.7. By Theorem 2.7, for any initial values
, constructing successively the sequences
,
,
, we have
,
as
. That is,
as . □
Theorem 3.3 Assume that with satisfies all the conditions of in Theorem 3.1
and satisfies (H
33)
and (H
34).
Then,
for any given ,
the integral equation (3.1)
has a unique positive solution in .
Moreover,
for any initial values ,
constructing successively the sequences:
we have , as . Further, the conclusions (i), (ii), and (iii) in Theorem 3.1 also hold.
Proof Similar to the proofs of Theorem 3.1 and Theorem 3.2, the conclusions follow from Theorem 2.12 and Corollary 2.14. □
3.2 Applications to nonlinear elliptic BVPs for the Lane-Emden-Fowler equations
Let Ω be a bounded domain with smooth boundary in
(
). Consider the following singular Dirichlet problem for the Lane-Emden-Fowler equation:
(3.3)
where and the nonlinear term is allowed to be singular on ∂ Ω.
The problem (3.3) arises in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrically materials (see [
28‐
32]). The theory of singular elliptic boundary value problems for partial differential equations has become an important area of investigation in the past three decades, see [
28‐
40] and references therein. By means of sub-supersolutions and various techniques related to the maximum principle for elliptic equations, some existence and nonexistence results, a unique positive solution are established. In [
3,
7], we investigated the existence and uniqueness of positive solutions to the singular Dirichlet problem for the Lane-Emden-Fowler equation (
3.3), where
is increasing in
for each
in [
3];
with
is increasing in
for each
,
in [
7] and
. However, to our knowledge, the results on the existence-uniqueness of positive solutions for singular elliptic equation are still few. The purpose here is to establish the existence-uniqueness of positive solutions to the singular Dirichlet problem for the Lane-Emden-Fowler equation (
3.3) by using some fixed point results in Section 2.
Throughout this subsection, denote
the Sobolev space (see [
41]), where
and
k is a nonnegative integer. And denote
the eigenfunction corresponding to the smallest eigenvalue
of the problem
in Ω, and
. For convenience, we can assume that
in
. Moreover, it is well known that (see for instance [
42]) there exist two positive constants
,
such that the first eigenvalue function satisfies
(3.4)
where .
Lemma 3.4 (See [[
43], Theorem 3, p.468])
Let Ω
be a bounded domain in with smooth boundary ∂ Ω.
Let and assume that,
for some ,
u satisfies
Then either , or there exists such that in Ω.
The proof of this result is due to Brezis and Nirenberg and the result is inspired by the work of Stampachia. Brezis and Nirenberg obtained this result in order to solve a similar eigenvalue problem as considered here. Actually, the result was extended to more general operators, such as
, under some suitable restrictions in order to solve a large class of problems (see for example the problems considered recently in the work of Covei [
44]). Here we recall the result since can be used to prove the following simple but useful lemma.
Lemma 3.5 (See [[
3], Theorem 3.1, p.1278])
Let Ω
be a bounded domain with smooth boundary in (
).
If and for ,
then there exists a constant such that
where depends only upon N and Ω.
Theorem 3.6
Assume that
and
(H35) , is nonnegative on , Hölder continuous in the variable x with the Hölder exponent for each and continuous in the variable u for each ;
(H
36)
is increasing in u for each x and is decreasing in u for each x,
and for any ,
there exists a constant ,
,
such that
(H
37)
,
satisfy the conditions of integrability,
i.
e.,
Then the problem (3.3) has a unique positive solution with respect to , where . Moreover, (i) if () for , then is strictly increasing in λ, that is, implies ; (ii) if there exists such that () for , then is continuous in λ, that is, () implies ; (iii) if there exists such that () for , then , .
Proof For the sake of convenience, set , the Banach space of continuous functions on with the norm . Set , the standard cone. It is clear that P is a normal cone in E and the normality constant is 1, is given as in the Section 1.1. We divide the proof into several steps.
Step 1. We consider the following linear elliptic boundary value problem:
(3.5)
where
. Since
, we can choose a sufficiently small number
such that
Then from (H
36),
(3.6)
(3.7)
Thus we get by applying the integrability condition (H
37) that says that
namely,
. By the classical theory of linear elliptic equations (see [
45]), the problem (3.5) admits a unique strong solution
. Recall that
. Using the Sobolev imbedding theory,
with
. Now we define an operator
by
where
is the unique strong solution of (3.5) for
. Evidently,
. Suppose that
ϕ is the solution of (3.5) with
, then
. Then from Lemma 3.5, there exists a positive constant
such that
(3.8)
Note that
. By the maximal principle,
. Since
for
, an application of Lemma 3.4 implies that
(3.9)
Combining (3.4) and (3.9), there exists a positive constant
such that
(3.10)
Let
is the unique strong solution of (3.5) for
. From (3.6) and (3.7), and applying the comparison principle, we conclude that
and, from (3.8) and (3.10), we get
for any
. So we find that
is well defined. Further, from (H
36) and the comparison principle, we can easily prove that
is increasing. In the following we prove that
for any
and
. For any
and
, we have
and
From (H
36) we also get
for any
. Therefore,
Using the comparison principle again, we can obtain immediately. So we have for , .
Step 2. We consider the following linear elliptic boundary value problem:
(3.11)
where
. Since
, we can choose a sufficiently small number
such that
Then from (H
36),
Thus we get by applying the integrability condition (H37) that says that , namely, . By the classical theory of linear elliptic equations, the problem (3.11) admits a unique strong solution . Recall that . Using the Sobolev imbedding theory, with . Now we define an operator by , , where is the unique strong solution of (3.11) for . Similar to Step 1,we can prove that is well defined. Using the comparison principle again, we can easily see that is decreasing and for any and .
Step 3. Now all the conditions of Corollary 2.2 are satisfied. It follows from Corollary 2.2 and Corollary 2.5 that the operator equation
has a unique solution
in
, that is,
. So
is a unique positive solution of the problem (3.3) in
for given
. By the theory of the linear elliptic equation, the problem
admits a unique solution , and hence . Recalling the uniqueness of the solution of (3.3), one can see easily that . Thus the problem (3.3) has a unique classical solution . Moreover, by using Corollary 2.5 and the theory of the linear elliptic equation, we can easily prove that (i) if () for , then is strictly increasing in λ, that is, implies ; (ii) if there exists such that () for , then is continuous in λ, that is, () implies ; (iii) if there exists such that () for , then , . □
Similar to the proofs of Theorem 3.6 and Theorems 3.2, 3.3, we can easily obtain the following conclusions.
Theorem 3.7 Assume that and satisfies all the conditions of Theorem 3.6, satisfies
(H38) is nonnegative on , Hölder continuous in the variable x with the Hölder exponent for each and is continuous in the variables u, v for each ;
(H
39)
for any ,
there exists a constant such that
(H
310)
satisfies the condition of integrability,
i.
e.,
Then the problem (3.3) has a unique positive solution with respect to , where . Further, the conclusions (i), (ii), and (iii) in Theorem 3.6 also hold.
Theorem 3.8 Assume that with satisfying all the conditions of in Theorem 3.6 and satisfying (H38), (H39), and (H310). Then the problem (3.3) has a unique positive solution with respect to , where . Further, the conclusions (i), (ii), and (iii) in Theorem 3.6 also hold.