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Zbl 1100.34019
Sun, Jingxian; Zhang, Guowei
Nontrivial solutions of singular superlinear Sturm--Liouville problems. (English)
[J] J. Math. Anal. Appl. 313, No. 2, 518-536 (2006). ISSN 0022-247X

The authors study the singular superlinear problem $$-(p(x)y')'-q(x)y=h(x)f(y),\quad 0<x<1,$$ $$\alpha_1 y(0)+\beta_1 y'(0)=0,\quad \alpha_2 y(1)+\beta_2 y'(1)=0.$$ The function $h$ is allowed to be singular at both $x=0$ and $x=1$. In addition, $f$ is not assumed to be nonnegative. The assumption of nonnegativity of $f$ has been very often required in the existing literature. Omitting this condition requires a different approach. Using topological degree theory, the authors establish conditions guaranteeing the existence of nontrivial solutions and positive solutions to the above boundary value problem. A nonsingular case is discussed as well.
[Pavel Rehak (Brno)]

MSC 2000:: *34B18 Positive solutions of nonlinear boundary value problems
34B16 Singular nonlinear boundary value problems

Keywords: singular Sturm-Liouville problem; nontrivial solution; positive solution; topological degree

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Zentralblatt MATH Berlin [Germany]