Existence of multiple positive solutions for Sturm-Liouville-like four-point boundary value problem with \(p\)-Laplacian. (English) Zbl 1145.34309

Ji, Dehong; Ge, Weigao

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 9, 2638-2646 (2008).

Summary: We consider the Sturm-Liouville-like four-point boundary value problem with \(p\)-Laplacian
\[ \begin{aligned} & (\phi_p(u'))'(t)+f(t,u(t))=0,\quad t\in (0,1),\\ & u(0)-\alpha u'(\xi)=0,\quad u(1)+\beta u'(\eta)=0,\end{aligned} \]
where \(\phi_p(s)=|s|^{p-2}s\), \(p>1\). By means of a fixed-point theorem for operators on a cone, the existence of multiple (at least three) positive solutions to the above boundary value problem is obtained.

MSC:

34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations

Keywords:

multiple positive solutions; Sturm-Liouville-like four-point boundary value problem; \(p\)-Laplacian operator

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