Abstract
We study
quasilinear elliptic equations of the type
-
Δ
p
u
=
σ
u
q
+
μ
{-\Delta_{p}u=\sigma u^{q}+\mu}
in
ℝ
n
{\mathbb{R}^{n}}
in the
case
0
<
q
<
p
-
1
{0<q<p-1}
, where μ and σ are nonnegative measurable functions, or locally finite measures, and
Δ
p
u
=
div
(
|
∇
u
|
p
-
2
∇
u
)
{\Delta_{p}u=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)}
is the p-Laplacian. Similar equations with
more general local and nonlocal operators in place of
Δ
p
{\Delta_{p}}
are treated as well.
We obtain existence criteria and global bilateral pointwise estimates for all positive solutions u:
u
(
x
)
≈
(
𝐖
p
σ
(
x
)
)
p
-
q
p
-
q
-
1
+
𝐊
p
,
q
σ
(
x
)
+
𝐖
p
μ
(
x
)
,
x
∈
ℝ
n
,
u(x)\approx({\mathbf{W}}_{p}\sigma(x))^{\frac{p-q}{p-q-1}}+{\mathbf{K}}_{p,q}%
\sigma(x)+{\mathbf{W}}_{p}\mu(x),\quad x\in\mathbb{R}^{n},
where
𝐖
p
{{\mathbf{W}}_{p}}
and
𝐊
p
,
q
{{\mathbf{K}}_{p,q}}
are, respectively, the Wolff potential and the intrinsic Wolff potential, with the constants of equivalence depending only on p, q, and n.
The contributions of μ and σ in these pointwise estimates
are totally separated, which
is a new phenomenon even when
p
=
2
{p=2}
.
Quasilinear elliptic equations with sub-natural growth terms in bounded domains
