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Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains

Vitali Liskevich·Sofya Lyakhova·Vitaly Moroz
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Abstract

We study the existence and nonexistence of positive (super) solutions to the nonlinear $p$-Laplace equation $$-\Delta_p u-\frac{\mu}{|x|^p}u^{p-1}=\frac{C}{|x|^{\sigma}}u^q$$ in exterior domains of ${\R}^N$ ($N\ge 2$). Here $p\in(1,+\infty)$ and $\mu\le C_H$, where $C_H$ is the critical Hardy constant. We provide a sharp characterization of the set of $(q,\sigma)\in\R^2$ such that the equation has no positive (super) solutions. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the $p$-Laplace operator with Hardy-type potentials, comparison principles and an improved version of Hardy's inequality in exterior domains. In the context of the $p$-Laplacian we establish the existence and asymptotic behavior of the harmonic functions by means of the generalized Pr\"ufer-Transformation.

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Vitali Liskevich, Sofya Lyakhova, Vitaly Moroz. 2006-04-21. Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains. https://doi.org/10.1016/j.jde.2006.09.001

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