Positive solutions of singular four-point boundary value problem with $p$-Laplacian

Chunmei Miao, College of Science, Changchun University, Changchun, P. R. China
Huihui Pang, College of Science, China Agricultural University, Beijing, P. R. China
Weigao Ge, Beijing Institute of Technology, Beijing, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 42. (2009), pp. 1-16.

Abstract: In this paper, we deal with the following singular four-point boundary value problem with $p$-Laplacian
$$
\left\{\begin{aligned}
&(\phi_{p}(u'(t)))'+q(t)f(t,u(t))=0,\ t\in(0,1),\\
&u(0)-\alpha u'(\xi)=0,\ u(1)+\beta u'(\eta)=0,
\end{aligned}\right.
$$
where $f(t,u)$ may be singular at $u=0$ and $q(t)$ may be singular at $t=0$ or $1$. By imposing some suitable conditions on the nonlinear term $f$, existence results of at least two positive solutions are obtained. The proof is based upon theory of Leray-Schauder degree and Krasnosel'skii's fixed point theorem.

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