Affiliation:
1. Graduate School of Mathematical Sciences, The University of Tokyo , 3-8-1 Komaba , Meguro-ku , Tokyo 153-8914 , Japan
Abstract
Abstract
This article is concerned with the structure of solutions to the elliptic problem for a Hénon-type equation with a forcing term:
−
Δ
u
=
α
(
x
)
u
p
+
κ
μ
,
in
R
N
,
u
>
0
,
in
R
N
,
(
P
κ
)
\hspace{11.3em}-\Delta u=\alpha \left(x){u}^{p}+\kappa \mu ,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}u\gt 0,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{13.0em}\left({{\rm{P}}}_{\kappa })
where
N
≥
3
N\ge 3
,
p
>
1
p\gt 1
,
κ
>
0
\kappa \gt 0
, and
α
\alpha
is a positive continuous function in
R
N
\
{
0
}
{{\mathbb{R}}}^{N}\setminus \left\{0\right\}
, and
μ
\mu
is a nonnegative Radon measure in
R
N
{{\mathbb{R}}}^{N}
. Under suitable assumptions on the exponent
p
p
, the coefficient
α
\alpha
, and the forcing term
μ
\mu
, we give a complete classification of the existence/nonexistence of solutions to problem (
P
κ
{{\rm{P}}}_{\kappa }
) with respect to
κ
\kappa
.