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Zbl 1125.34018
Zhang, Xinguang; Liu, Lishan
Positive solutions of fourth-order four-point boundary value problems with $p$-Laplacian operator. (English)
[J] J. Math. Anal. Appl. 336, No. 2, 1414-1423 (2007). ISSN 0022-247X

Summary: We consider the existence of positive solutions for the singular fourth-order $p$-Laplacian equation $$[\varphi_p(u''(t))]''=f(t,u(t)),\ 0<t<1,$$ with the four-point boundary conditions $$u(0)=0,\quad u(1)= au(\xi),\quad u''(0)=0,\quad u''(1)=bu''(\eta),$$ where $\varphi_p(t)= |t|^{p-2}t$, $p>1$, $0<\xi$, $\eta<1$, $f\in C((0,1)\times(0,+\infty)$, $[0,+\infty))$ may be singular at $t=0$ and/or 1 and $u=0$. By using the upper and lower solution method and fixed-point theorems, the existence of positive solutions to the above the boundary value problem is obtained.

MSC 2000:: *34B18 Positive solutions of nonlinear boundary value problems
34B16 Singular nonlinear boundary value problems
34B10 Multipoint boundary value problems

Keywords: $p$-Laplacian operator; singular boundary value problems; fixed-point theorem; positive solution

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