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Zbl 1111.34023
Sun, Jingxian; Zhang, Guowei
Nontrivial solutions of singular sublinear Sturm--Liouville problems.
(English)
[J] J. Math. Anal. Appl. 326, No. 1, 242-251 (2007). ISSN 0022-247X

Summary: The singular sublinear Sturm-Liouville problems $$\cases -(L\varphi) (x)=h(x)f(\varphi(x)),\quad 0<x<1,\\ R_1(\varphi)=\alpha_1 \varphi(0)+ \beta_1 \varphi'(0)=0,\quad R_2(\varphi)=\alpha_2\varphi(1)+ \beta_2 \varphi'(1)=0, \endcases$$ are considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators, where $(L\varphi)(x)=(p(x)\varphi'(x))'+q(x)\varphi(x)$ 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 on nontrivial solutions and positive solutions are given by means of the topological degree theory.
MSC 2000:
*34B24 Sturm-Liouville theory
34B16 Singular nonlinear boundary value problems
34B18 Positive solutions of nonlinear boundary value problems

Keywords: second-order singular equation; two-point boundary value problem; nontrivial solution; positive solution

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