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

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 ${\mathbb{R}}^N$ ($N\ge 2$). 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 $(q,\sigma )\in {\mathbb{R}}^2$ 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 Prüfer-Transformation.

34 pages, 1 figure

References in corpus (2)

• Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains
• A critical phenomenon for sublinear elliptic equations in cone-like domains

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