Nonlinear Analysis: Theory, Methods & Applications
Existence of nodal solutions of a nonlinear eigenvalue problem with indefinite weight function
Section snippets
Introduction and the main result
The nonlinear eigenvalue problem has been studied by many authors, where satisfies:
(A) is continuous and does not vanish identically on any subinterval of .
In [1], Henderson and Wang were concerned with determining values of for which there exist positive solutions of (1.1) under condition (A) and some suitable conditions on . Recently in [2], Ma and Thompson were concerned with determining values of for which there exist nodal
Proof of the main result
Let with the norm Let with the norm Define by setting where Then is compact.
Letting be such that
then clearly Letting then is nondecreasing and
Proof of Theorem 1.1 We divide the proof into two cases: and . Case 1:. Let us consider
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. The first author was supported by NSFC (No. 10671158), the NSF of Gansu Province (No. 3ZS051-A25-016), NWNU-KJCXGC-03-17, the Spring-sun program (No. Z2004-1-62033), SRFDP (No. 20060736001), and the SRF for ROCS, SEM (2006[311]).
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