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

An Upper and Lower Solution Method for the One-Dimensional Singular $p$-Laplacian

abstract:
The singular boundary value problem
\[\left\{\begin{array}{l}
(\varphi _p(y^{\prime }))^{\prime }+q\left( t\right) f(t,y)=0,\;\hbox{ for}\;t\in (0,1), \\
y(0)=y(1)=0
\end{array}\right.\]
is studied in this paper with $\varphi _p(s)=\left| s\right| ^{p-2}s$, $p>1$. The nonlinearity may be singular at $y=0,t=0$ and $t=1,$ and the function $f$ may change sign. An upper and lower solution approach is presented.

Mathematics Subject Classification: 34B15.

Key words and phrases: Positive solution, boundary value problem, upper and lower solutions.