Nonresonance for a one-dimensional $p$-Laplacian with strong singularity

Abstract

In this work, we give nonresonance conditions for a singular quasilinear two-point boundary value problem

$$\{\begin{array}{cc}{\left({\phi }_p\left(u^{\prime }\right)\right)}^{\prime }+h\left(t\right)f(t,u)=k(t,u,u^{\prime })\text{,}& \text{a.e. in}(0,1)\text{,}\\ u\left(0\right)=u\left(1\right)=0\text{,}\end{array}$$

where ${\phi }_p\left(x\right)={\left|x\right|}^{p-2}x,p>1$,$f\in C([0,1]\times \mathbb{R},\mathbb{R})$,$h$ is a nonnegative measurable function on $(0,1)$, and $k:(0,1)\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function dominated by $K\in L^1(0,1)$, i.e., $\left|k(t,x,y)\right|\le K\left(t\right)$ for all $(t,x,y)\in (0,1)\times \mathbb{R}\times \mathbb{R}$.