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 d i v ( | x | α | ∇ u | p − 2 ∇ u ) + μ | x | p − α | u | p − 2 u = 0 in R N − { 0 } \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 ∈ N N\in {\mathbf {N}} , p > 1 p>1 and α ∈ R \alpha \in {\mathbf {R}} , μ ∈ R − { 0 } \mu \in {\mathbf {R}}-\{0\} . In this note, the asymptotic behaviors of radial solutions are obtained divided into three case μ > | ( N − p + α ) / p | p \mu >|(N-p+\alpha )/p|^p , μ = | ( N − p + α ) / p | p \mu =|(N-p+\alpha )/p|^p and μ > | ( N − p + α ) / p | p \mu >|(N-p+\alpha )/p|^p . This equation also appears as the Euler-Lagrange equation related to the weighted Hardy inequality ∫ Ω | ∇ u ( x ) | p | x | α d x ≥ | N − p + α p | p ∫ Ω | u ( x ) | p | x | α − p d x \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 ∈ C c ∞ ( R N ) u \in C_c^\infty ({\mathbf {R}}^N) and N − p + α ≠ 0 N-p+\alpha \ne 0 , where Ω \Omega is a domain of R N {\mathbf {R}}^N . The rectifiability of oscillatory solutions to the ordinary differential equation with one-dimensional p p -Laplacian is also studied, and an answer to an open problem is given.