Global Structure of Nodal Solutions for Second-Order m-Point Boundary Value Problems with Superlinear Nonlinearities
© Yulian An. 2011
Received: 8 May 2010
Accepted: 23 September 2010
Published: 28 September 2010
In the case that the global structure of nodal solutions of nonlinear second-order -point eigenvalue problems (1.1), (1.2) have been extensively studied; see [1–5] and the references therein. However, relatively little is known about the global structure of solutions in the case that and few global results were found in the available literature when The likely reason is that the global bifurcation techniques cannot be used directly in the case. On the other hand, when -point boundary value condition (1.2) is concerned, the discussion is more difficult since the problem is nonsymmetric and the corresponding operator is disconjugate. In , we discussed the global structure of positive solutions of (1.1), (1.2) with However, to the best of our knowledge, there is no paper to discuss the global structure of nodal solutions of (1.1), (1.2) with
In this paper, we obtain a complete description of the global structure of nodal solutions of (1.1), (1.2) under the following assumptions:
The nodal properties of solutions of nonlinear Sturm-Liouville problems with separated boundary conditions are usually described in terms of sets similar to see . However, Rynne  stated that are more appropriate than when the multipoint boundary condition (1.2) is considered.
Let under the product topology. As in , we add the points to our space Let Let denote the closure of set of those solutions of (1.1), (1.2) which belong to The main results of this paper are the following.
Let (A1)–(A4) hold.
Let (A1)–(A4) hold.
The rest of the paper is organized as follows. Section 2 contains some preliminary propositions. In Section 3, we use the global bifurcation theorems to analyse the global behavior of the components of nodal solutions of (1.1), (1.2).
Definition 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
It is easy to verify that the following lemma holds.
The proof is similar to that of Lemma in ; we omit it.
3. Proof of the Main Results
Lemma 3.1 (see [1, Proposition ]).
The results of Rabinowitz  for (3.8) can be stated as follows. For each integer , there exists a continuum of solutions of (3.8) joining to infinity in Moreover,
Proof of Theorem 1.5.
has a unique solution on for every and . Therefore, any nontrivial solution of (1.1), (1.2) has only simple zeros in and . Meanwhile, (A1) implies that [1, proposition 4.1]. Since , we conclude that . Moreover, by (1.1) and (1.2).
We divide the proof into three cases.
In this case, we can show easily that joins with by using the same method used to prove Theorem in .
denote the zeros of in . Then, after taking a subsequence if necessary, . Clearly, . Set . We can choose at least one subinterval which is of length at least for some . Then, for this if is large enough. Put .
We see that consists of a sequence of positive and negative bumps, together with a truncated bump at the right end of the interval with the following properties (ignoring the truncated bump) (see, ):
(i)all the positive (resp., negative) bumps have the same shape (the shapes of the positive and negative bumps may be different);
(iii)all the positive (negative) bumps attain the same maximum (minimum) value.
Proof of Theorem 1.6.
The author is very grateful to the anonymous referees for their valuable suggestions. This paper was supported by NSFC (no.10671158), 11YZ225, YJ2009-16 (no.A06/1020K096019).
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