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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Solutions in Lebesgue spaces to nonlinear elliptic equations with subnatural growth terms

Authors: A. Seesanea and I. E. Verbitsky
Original publication: Algebra i Analiz, tom 31 (2019), nomer 3.
Journal: St. Petersburg Math. J. 31 (2020), 557-572
MSC (2010): Primary 35J92; Secondary 35J20, 42B37
Published electronically: April 30, 2020
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Abstract: The paper is devoted to the existence problem for positive solutions $ {u \in L^{r}(\mathbb{R}^{n})}$, $ 0<r<\infty $, to the quasilinear elliptic equation

$\displaystyle -\Delta _{p} u = \sigma u^{q} \ $$\displaystyle \text { in } \ \mathbb{R}^n$    

in the subnatural growth case $ 0<q< p-1$, where $ \Delta _{p}u = \mathrm {div}( \vert\nabla u\vert^{p-2} \nabla u )$ is the $ p$-Laplacian with $ 1<p<\infty $, and $ \sigma $ is a nonnegative measurable function (or measure) on $ \mathbb{R}^n$.

The techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic operators in place of $ \Delta _{p}$ such as the $ \mathcal {A}$-Laplacian $ \mathrm {div} \mathcal {A}(x,\nabla u)$, or the fractional Laplacian $ (-\Delta )^{\alpha }$ on $ \mathbb{R}^n$, as well as linear uniformly elliptic operators with bounded measurable coefficients $ \mathrm {div} (\mathcal {A} \nabla u)$ on an arbitrary domain $ \Omega \subseteq \mathbb{R}^n$ with a positive Green function.

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Additional Information

A. Seesanea
Affiliation: Department of Mathematics, Hokkaido University, Sapporo, Hokkaido 060-0810, Japan

I. E. Verbitsky
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Keywords: Quasilinear elliptic equation, measure data, $p$-Laplacian, fractional Laplacian, Wolff potential, Green function
Received by editor(s): November 1, 2018
Published electronically: April 30, 2020
Additional Notes: A. S. is partially supported by JSPS KAKENHI Grant no. 17H01092
Dedicated: Dedicated to the memory of S. G. Mikhlin
Article copyright: © Copyright 2020 American Mathematical Society