Abstract
We establish a 3G-Theorem for the Green's
function for an unbounded regular domain D in
ℝn(n ≥ 3), with compact boundary.
We exploit this result to introduce a new class of potentials
K(D) that properly contains the classical Kato class
$K_n^\infty(D)$
.
Next, we study the existence and the uniqueness of a positive
continuous solution u in
$\bar{D}$
of the following nonlinear singular elliptic
problem
\[
\left\{ \begin{array}{ll}
\Delta u +\varphi (.,u) =0\,, \quad &\mbox{in $D$ (in the sense of
distributions)}\\[3pt]
u=0\quad &\mbox{on}\ \partial D\\[3pt]
u(x)\rightarrow 0\,, \quad &\mbox{as}\ |x|\rightarrow\infty\,,
\end{array}\right.
\]
where φ is a nonnegative Borel measurable function in
D × (0, ∞), that belongs to a convex cone
which contains, in particular, all functions
φ(x, t) = q(x)t-σ,
σ ≥ 0 with q ∈ K(D). We give also
some estimates on the solution u.