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Existence of positive solutions for nonlinear singular boundary value problem with \(p\)-Laplacian. (English) Zbl 1188.34030

Summary: We study the existence of positive solutions for the following Sturm-Liouville-like four-point singular boundary value problem (BVP) with \(p\)-Laplacian
\[ \phi_p(u'(t)))'+q(t)f(u(t))=0,\quad t\in (0,1), \]
\[ u(0)-\alpha u'(\xi)=0,\quad u(1)+\beta u(\eta)=0 \]
where \(\phi_p(s)=|s|^{p-2}s\), \(p>1\), \(f\) is a lower semi-continuous function. Using the fixed-point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for the Sturm-Liouville-like singular BVP with \(p\)-Laplacian is obtained.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory
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