Ravi P. Agarwal, Haishen Lü and Donal O'Regan

Positive Solutions for the Boundary Value Problem
$(| u''|^{p-2}u'')''-\lambda q(t)f(u(t))=0$

abstract:
This paper considers the boundary value problem:
\[
\left\{\begin{array}{l}
(| u''| ^{p-2}u'') ''-\lambda q(t)f(u( t) )=0,\;\hbox{in }(0,1) , \\
u(0)=u(1)=u''t( 0) =u''(1) =0,
\end{array}\right.
\]
with l >0. The value of  l is chosen so that the boundary value problem has a positive solution. Moreover, we derive an explicit interval for l such that for any l in this interval, the existence of a positive solution to the boundary value problem is guaranteed. In addition we also discuss the existence of two positive solutions for l in an appropriate interval.

Mathematics Subject Classification: 34B15.

Key words and phrases: Boundary value problem, Positive solution, Beam equation.