Existence of Positive Solutions of a Singular Nonlinear Boundary Value Problem
© R. Ma and J. Li. 2010
Received: 21 May 2010
Accepted: 11 August 2010
Published: 18 August 2010
We are concerned with the existence of positive solutions of singular second-order boundary value problem , , , which is not necessarily linearizable. Here, nonlinearity is allowed to have singularities at . The proof of our main result is based upon topological degree theory and global bifurcation techniques.
by applying Rabinowitz's global bifurcation theorem, where is allowed to have singularities at and is linearizable at as well as at . It is the purpose of this paper to study the existence of positive solutions of (1.1), which is not necessarily linearizable.
A function is said to be an -Carathéodory function if it satisfies the following:
(i)for each , is measurable;
(ii)for a.e. , is continuous;
(iii)for any , there exists such that
In this paper, we will prove the existence of positive solutions of (1.1) by using the global bifurcation techniques under the following assumptions.
(H1) Let be an -Carathéodory function and there exist functions , , , and such that
uniformly for a.e. .
(H2) for a.e. and .
The main tool we will use is the following global bifurcation theorem for problem which is not necessarily linearizable.
Theorem A (Rabinowitz, ).
then there exists a continuum (i.e., a closed connected set) of containing , and either
(i) is unbounded in , or
To state our main results, we need the following.
Lemma 1.3 (see [1, Proposition ]).
Moreover, for each , is simple and its eigenfunction has exactly zeros in .
Our main result is the following.
then (1.1) has at least one positive solution.
2. Preliminary Results
Lemma 2.1 (see [15, Proposition ]).
Then, from Lemma 2.1, is well defined.
This completes the proof.
Lemma 2.4 (see [1, Lemma ]).
is precompact in .
Since for a.e. , Lemma 2.2 yields for . Thus, is a nonnegative solution of (2.19), and the closure of the set of nontrivial solutions of (2.21) in is exactly .
Moreover, , whenever .
Let (H1) and (H2) hold. Then the operator is completely continuous.
By the Lebesgue dominated convergence theorem, we have that in as . Thus, is continuous.
Let be a bounded set in . Lemma 2.4 together with (2.28) shows that is precompact in . Therefore, is completely continuous.
For , let , let denote the degree of on with respect to .
Suppose to the contrary that there exist sequences and in in , such that for all , then, in .
Thus, . This contradicts .
For and , .
which ends the proof.
where is the nonnegative eigenfunction corresponding to .
where and . Since on and , we have from (2.46) that .
This contradicts (2.47).
For and , .
Now, using Theorem A, we may prove the following.
For fixed with , let us take that , and . It is easy to check that, for , all of the conditions of Theorem A are satisfied. So there exists a connected component of solutions of (2.30) containing , and either
(i) is unbounded, or
By Lemma 2.7, the case (ii) can not occur. Thus, is unbounded bifurcated from in . Furthermore, we have from Lemma 2.7 that for any closed interval , if , then in is impossible. So must be bifurcated from in .
3. Proof of the Main Results
Proof of Theorem 1.5.
We note that for all since is the only solution of (2.30) for and .
We divide the proof into two steps.
We show that is bounded.
Let denote the nonnegative eigenfunction corresponding to .
We show that joins to .
So joins to .
Again joins to and the result follows.
under the following assumptions:
satisfying has at least one zero in and has no zeros in .
It is worth remarking that (A1)-(A2) imply Condition (1.21) in Theorem 1.5. However, Condition (1.21) is easier to be verified than (A1)-(A2) since and are easily estimated by Rayleigh's Quotient.
The language of eigenvalue of singular linear eigenvalue problem did not occur until Asakawa  in 2001. The first part of Theorem 1.5 is new.
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC 11061030, the Fundamental Research Funds for the Gansu Universities.
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