Multiple existence of positive even solutions for a two point boundary value problem on some very narrow possible parameter set

Abstract

The existence of multiple positive even solutions to{u″(x)+(|x|l+λ)u(x)p=0,x∈(−1,1),u(−1)=u(1)=0, is proved, where the parameters l and λ satisfy l≥0 and λ≥0 and the exponent p satisfies p>1. It is shown that for fixed p>1, on the majority part of the first quadrant of (l,λ)⊂R2, the uniqueness of positive even solutions of the above problem holds and a very narrow set remains as the possible region of the existence of multiple positive even solutions. Thus, it seems natural to expect the uniqueness of positive even solutions holds on the whole set of the first quadrant of (l,λ)⊂R2. Contrary to this expectation, we find some triples (l,λ,p) in such a narrow set such that positive even solutions exist multiply with the aid of the numerical verification method.