Large time behaviour of solutions of a generalized Haraux-Weissler equation

Abstract

We characterize the possible large time behaviors of solutions to $$ (|U|^{m-1} U)'' + {{N-1}\over\eta } (|U|^{m-1} U)' + \beta\eta U' + \alpha U + \gamma |U|^{p-1} U \, =\, 0 \quad \text{ in }\ \mathbb{R}^+ $$ with $m>0$, $p>1$, $\alpha, \beta >0$, $\alpha (m-1)+2\beta >0$ and $\gamma \in \{ -1,0,1\}$. It turns out that if $U$ is bounded and non-constant, $L:= \lim_{\eta\rightarrow\infty } \eta^{\alpha /\beta } U(\eta )$ is always finite and that in the case in which $L=0$, the solutions have compact support ($m>1$), decay exponentially ($m=1$) or decay like $\eta ^{-{2\over{1-m}}}$ ($m<1$), respectively. We want to stress that we impose no sign restriction on the solution.