Asymptotics of Solutions to $p$-Laplacian Equations Involving Convection and Reaction Terms

Resumo

The purpose of this work is to investigate a nonlinear $p$-Laplacian equation that incorporates both convection and reaction effects. The model under consideration takes the form
$$
\displaystyle \mbox{div}(|\nabla U|^{p-2} \nabla U) + \lambda x\nabla(|U|^{q-1} U) + \theta U = 0 \quad \mbox{in} \quad \mathbb{R}^{N}, \\
%\displaystyle \left( |u'|^{p-2} u' \right)' + \frac{N-1}{r} |u'|^{p-2} u' + \lambda r(|u|^{q-1} u)' + \theta u = 0, \quad r > 0,
$$
with parameters $N \geq 1$, $p>2$, $q>1$, $\lambda>0$, and $\theta>0$.
Our main results concern the existence of global radial solutions, which are shown to be strictly positive under suitable assumptions. In addition, we examine the qualitative properties of these solutions and describe their asymptotic profile as $|x|\rightarrow\infty$.