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

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