On unbounded solutions of singular IVPs with $\phi$-Laplacian

Abstract

The paper deals with a singular nonlinear initial value problem with a $\phi$-Laplacian $$ (p(t)\phi(u'(t)))'+ p(t)f(\phi(u(t)))=0,\ t>0, \quad u(0)=u_0 \in [L_0,L],\ u'(0)=0. $$ Here, $f$ is a continuous function with three roots $\phi(L_0)<0<\phi(L)$, $\phi:\mathbb{R} \to\mathbb{R}$ is an increasing homeomorphism and function $p$ is positive and increasing on $(0,\infty)$. The problem is singular in the sense that $p(0)=0$ and $1/p$ may not be integrable in a neighbourhood of the origin. The goal of this paper is to prove the existence of unbounded solutions. The investigation is held in two different ways according to the Lipschitz continuity of functions $\phi^{-1}$ and $f$. The case when those functions are not Lipschitz continuous is more involved that the opposite case and it is managed by means of the lower and upper functions method. In both cases, existence criteria for unbounded solutions are derived.

Keywords

• second order ode
• time singularity
• $\phi$-laplacian
• unbounded solution
• escape solution
• lower and upper functions method