Nontrivial solutions of singular sublinear Sturm–Liouville problems☆

By Jingxian Sun, Guowei Zhang. 2006

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Abstract The singular superlinear Sturm–Liouville problems { − ( L φ ) ( x ) = h ( x ) f ( φ ( x ) ) , 0 x 1 , R 1 ( φ ) = α 1 φ ( 0 ) + β 1 φ ′ ( 0 ) = 0 , R 2 ( φ ) = α 2 φ ( 1 ) + β 2 φ ′ ( 1 ) = 0 , are considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators, where ( L φ ) ( x ) = ( p ( x ) φ ′ ( x ) ) ′ + q ( x ) φ ( x ) and h ( x ) is allowed to be singular at both x = 0 and x = 1 . In particular, f is not necessary to be nonnegative. The existence results of nontrivial solutions and positive solutions are given by means of the topological degree theory.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0e95326bdb8cf9915d719783f563aeeda92f7b77