We consider the boundary Hardy–Hénon equation
\[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \]
where
$B_1(0)\subset \mathbb {R}^{N}$
$(N\geq 3)$
is a ball of radial
$1$
centred at
$0$
,
$p>0$
and
$\alpha \in \mathbb {R}$
. We are concerned with the estimate, existence and nonexistence of positive solutions of the equation, in particular, the equation with Dirichlet boundary condition. For the case
$0< p<({N+2})/({N-2})$
, we establish the estimate of positive solutions. When
$\alpha \leq -2$
and
$p>1$
, we give some conclusions with respect to nonexistence. When
$\alpha >-2$
and
$1< p<({N+2})/({N-2})$
, we obtain the existence of positive solution for the corresponding Dirichlet problem. When
$0< p\leq 1$
and
$\alpha \leq -2$
, we show the nonexistence of positive solutions. When
$0< p<1$
,
$\alpha >-2$
, we give some results with respect to existence and uniqueness of positive solutions.
Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case
