Эта публикация цитируется в
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Статьи
Solutions in Lebesgue spaces to nonlinear elliptic equations with subnatural growth terms
A. Seesaneaa,
I. E. Verbitskyb a Department of Mathematics, Hokkaido University, Sapporo, Hokkaido 060-0810, Japan
b Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA
Аннотация:
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
$$
-\Delta _{p} u = \sigma u^{q} \text { in } \mathbb{R}^n
$$
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.
Ключевые слова:
quasilinear elliptic equation, measure data,
$p$-Laplacian, fractional Laplacian, Wolff potential, Green function.
MSC: Primary
35J92; Secondary
35J20,
42B37 Поступила в редакцию: 01.11.2018
Язык публикации: английский