Asymptotic Profiles of Global Solutions for a PDE Including a Singular Nonlinearity

Abstract

We examine solutions of global nature to the following elliptic equation with singular nonlinearity
\[
\Delta_p U + \lambda_1 U + \lambda_2 |x|^{\gamma} |U|^{q-1}U = 0, \quad \text{in } \mathbb{R}^N,
\]
such that $p > 2$, $q \geq 1$, $N \geq 1$, $\lambda_1 > 0$, $\lambda_2> 0$, and $\gamma < 0$. \\
We prove the global existence of radial solutions and examine conditions ensuring their strict positivity. Furthermore, we describe the asymptotic profile of positive solutions in the limit near infinity.