Periodic singular problem with quasilinear differential operator

Abstract

We study the singular periodic boundary value problem of the form \[ \left(\phi (u^{\prime })\right)^{\prime }+h(u)u^{\prime }=g(u)+e(t),\quad u(0)=u(T),\quad u^{\prime }(0)=u^{\prime }(T), \] where $\phi \:\mathbb{R}\rightarrow \mathbb{R}$ is an increasing and odd homeomorphism such that $\phi (\mathbb{R})=\mathbb{R},$ $h\in C[0,\infty ),$ $e\in L_1J$ and $g\in C(0,\infty )$ can have a space singularity at $x=0,$ i.e. $\limsup _{x\rightarrow 0+}|g(x)|=\infty $ may hold. We prove new existence results both for the case of an attractive singularity, when $\liminf _{x\rightarrow 0+}g(x)=-\infty ,$ and for the case of a strong repulsive singularity, when $\lim _{x\rightarrow 0+}\int _x^1g(\xi )\hspace{0.56905pt}\text{d}\xi =\infty .$ In the latter case we assume that $\phi (y)=\phi _p(y)=|y|^{p-2}y,$ $p>1,$ is the well-known $p$-Laplacian. Our results extend and complete those obtained recently by Jebelean and Mawhin and by Liu Bing.