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The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations

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Abstract

We investigate the boundary growth of positive superharmonic functions u on a bounded domain Ω in \(\mathbb {R}^{n}\) , n ≥ 3, satisfying the nonlinear elliptic inequality

$$0 \le - \Delta u \le c\delta_{\Omega}(x)^{-\alpha}u^p \quad {\rm in}\ \Omega,$$

where c >  0, α ≥ 0 and p >  0 are constants, and \(\delta_{\Omega}(x)\) is the distance from x to the boundary of Ω. The result is applied to show a Harnack inequality for such superharmonic functions. Also, we study the existence of positive solutions, with singularity on the boundary, of the nonlinear elliptic equation

$$-\Delta u + Vu = f(x, u) \quad {\rm in} \ \Omega,$$

where V and f are Borel measurable functions conditioned by the generalized Kato class.

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  1. Faculty of Education and Human Studies, Akita University, Akita, 010-8502, Japan

    Kentaro Hirata

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  1. Kentaro Hirata
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Hirata, K. The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations. Math. Ann. 340, 625–645 (2008). https://doi.org/10.1007/s00208-007-0163-6

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