New embeddings of weighted Sobolev spaces are established. Using such embeddings, we obtain the existence and regularity of positive solutions with Navier boundary value problems for a weighted fourth order elliptic equation. We also obtain Liouville type results for the related equation. Some problems are still open.
We generalize the Bôcher-type theorem and give a sharp characterization of the behavior at the isolated singularities of a solution bounded on one side for the equation
Δ
g
u
=
0
\Delta _g u =0
on singular manifolds with conical metrics. Furthermore, we also obtain a Liouville-type result which demonstrates that the fundamental solution is the unique nontrivial solution of
div
(
|
x
|
θ
∇
u
)
=
0
\operatorname {div}(|x|^\theta \nabla u)=0
in
R
n
∖
{
0
}
\mathbb {R}^n\setminus \{0\}
that is bounded on one side in both a neighborhood of the origin as well as at infinity.
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