Let
$M$ be a complete Riemannian
manifold and
$G$ a Lie subgroup
of the isometry group of
$M$
acting freely and properly on
$M$.
We study the Dirichlet problem
$$\left\{\begin{array}{c}\phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \text{div}\left(\frac{a\left(\parallel \nabla u\parallel \right)}{\parallel \nabla u\parallel}\nabla u\right)=0,\phantom{\rule{1em}{0ex}}\hfill & \text{in}\Omega ,\hfill \\ u\partial \Omega =\phi ,\phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \end{array}\right.$$
where
$\Omega $ is a
$G$invariant
domain of
${C}^{2,\alpha}$class
in
$M$ and
$\phi \in {C}^{2,\alpha}\left(\partial \overline{\Omega}\right)$ is a
$G$invariant
function. Two classical PDEs are included in this family: the
$p$Laplacian
($a\left(s\right)={s}^{p1}$,
$p>1$) and the minimal
surface equation ($a\left(s\right)=s\u2215\sqrt{1+{s}^{2}}\phantom{\rule{0.3em}{0ex}}$).
Our motivation, by using the concept of Riemannian submersion, is to present a method for
studying
$G$invariant
solutions for noncompact Lie groups which allows the reduction of the Dirichlet
problem on unbounded domains to one on bounded domains.
