A note on the asymptotic behavior of radial solutions to quasilinear elliptic equations with a Hardy potential

Abstract:

The quasilinear elliptic equation with a Hardy potential \begin{equation*} {\mathrm {div}}(|x|^\alpha |\nabla u|^{p-2}\nabla u) + \frac {\mu }{|x|^{p-\alpha }}|u|^{p-2}u = 0 \quad \text {in} \ {\mathbf {R}}^N-\{0\} \end{equation*} is considered, where $N\in {\mathbf {N}}$, $p>1$ and $\alpha \in {\mathbf {R}}$, $\mu \in {\mathbf {R}}-\{0\}$. In this note, the asymptotic behaviors of radial solutions are obtained divided into three case $\mu <|(N-p+\alpha )/p|^p$, $\mu =|(N-p+\alpha )/p|^p$ and $\mu >|(N-p+\alpha )/p|^p$. This equation also appears as the Euler-Lagrange equation related to the weighted Hardy inequality \begin{equation*} \int _\Omega |\nabla u(x)|^p |x|^\alpha dx \ge \Biggl | \frac {N-p+\alpha }{p} \Biggr |^p \int _\Omega |u(x)|^p |x|^{\alpha -p} dx \end{equation*} for $u \in C_c^\infty ({\mathbf {R}}^N)$ and $N-p+\alpha \ne 0$, where $\Omega$ is a domain of ${\mathbf {R}}^N$.

The rectifiability of oscillatory solutions to the ordinary differential equation with one-dimensional $p$-Laplacian is also studied, and an answer to an open problem is given.