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

Nonuniform Nonresonance at the First Eigenvalue of the One-Dimensional Singular p-Laplacian

abstract:
In this paper, general existence theorems are presented for the singular equation
\[
\begin{cases}
-\left( \varphi _p\left( u^{\prime }\right) \right) ^{\prime}=f\left(t,u,u^{\prime }\right) ,\;\;0<t<1, \\
u\left( 0\right) =u\left( 1\right) =0.
\end{cases}
\]
Throughout, our nonlinearity is allowed to change sign. The singularity may occur at $u=0,$ $t=0$, $t=1$ and $f\,$ may be nonuniform nonresonant at the first eigenvalue.

Mathematics Subject Classification: 34B15, 34B16

Key words and phrases: One dimensional singular p-Laplacian, positive solution, nonuniform nonresonance condition.