Homoclinic solutions of singular differential equations with $\phi$-Laplacian

Abstract

A singular nonlinear initial value problem (IVP) with a $\phi$-Laplacian of the form $$ (p(t)\phi(u'(t)))'+ p(t)f(\phi(u(t)))=0, \quad u(0)=u_0 \in [L_0,0),\quad u'(0)=0 $$ is investigated on the half-line $[0,\infty)$. Here, function $\phi$ is smooth and increasing on $\mathbb{R}$ with $\phi(0)=0$, function $f$ is locally Lipschitz continuous with three zeros $\phi(L_0)<0<\phi(L)$, function $p$ is smooth and increasing on $(0,\infty)$, and the problem is singular in the sense that $p(0)=0$ and $1/p(t)$ may not be integrable on $[0,1]$. The main result of the paper is the existence of homoclinic solutions defined as nondecreasing solutions $u$ of the IVP satisfying $\lim_{t\to \infty}u(t)=L$.

Keywords

• second order ode
• time singularity
• $\phi$-laplacian
• homoclinic solution
• half-line