2$. Let
$$v_p=Du_p \cdot x=\sum_{i=1}^n x_iu_{p,x_i},$$
$$A^p=(A^p_{ij})_{i,j=1,...,n},$$
$$(A^p_{ij})=|Du_p|^{p-2}\delta _{ij}+(p-2)|Du_p|
^{p-4}u_{p,x_i}u_{p,x_j}.$$
Then $v_p$ satisfies the following identity
$$\int_\Omega A^pDv_p\cdot D\phi =0,\qquad
\forall\phi\in C^\infty_0(\Omega).\eqno(2)$$
\noindent{\bf Proof.} For simplicity of notations we shall
drop the index $p$ from $A^p$, $u_p$, $v_p$.
Let us recall that, by a well known result of K. Uhlenbeck ([U]),
weak solutions to $(1_p)$ belong to
$C^{1,\alpha}_{loc}(\Omega)$
and satisfy
$|Du|^{{p-2 \over 2}}u_{x_ix_j}\in
L^2_{loc}(\Omega)$, so that $ADv\in L^2_{loc}(\Omega)$.
Let $\phi\in C^{\infty}_0(\Omega)$. Then, by equation $(1_p)$,
$$\int_\Omega ADv\cdot D\phi=
\int_\Omega |Du|^{p-2}x_iDu_{x_i}\cdot D\phi
+\int_\Omega |Du|^{p-2}Du\cdot D\phi+$$
$$+\int_\Omega (p-2)|Du|^{p-4}x_iu_{x_j}u_{x_ix_j}
Du\cdot D\phi
+\int_\Omega (p-2)|Du|^{p-4}|Du|^2
Du\cdot D\phi=$$
$$=
\int_\Omega x_i\left(|Du|^{p-2}Du\right)_{x_i}
\cdot D\phi=
-\int_\Omega |Du|^{p-2}Du\cdot
\left(\sum_i (x_iD\phi)_{x_i}\right)$$
$$=-\int_\Omega |Du|^{p-2}Du\cdot Dh=0,$$
where $h=\sum_{i=1}^n x_i\phi_{x_i}+(n-1)\phi$. {\tenmsa \char003}
\medskip
\noindent
{\bf Remark.} Let us point out that (2) means that
$v_p$ is a distributional solution to the following degenerate
elliptic equation
$$\hbox{div}(|Du_p|^{p-2}Dv_p
+(p-2)|Du_p|^{p-4}Du_p \cdot Dv_pDu_p)=0.$$
The degeneracy of this equation prevents to replace
$C^\infty_0$ with $W^{1,2}_0$ test functions and
therefore a maximum principle is not straightforward.
The suitable maximum principle will be derived in the
proof of Theorem 1, under hypotheses which ensure
that a Hopf-type Lemma holds.
\proclaim
Lemma 2 (A uniform Hopf-type Lemma).
Let $\Omega=\Omega_2\setminus
\bar\Omega_1$, where
$\Omega_1$ and $\Omega_2$,
$\Omega_1\subset \subset\Omega_2$ are two simply
connected bounded domains in $\Bbb R^n$
with $C^2$ boundaries $\partial\Omega_1$,
$\partial\Omega_2$. Let $u_p$ be the solution
to $(1_p)$-(BC).
Then there exist a constant $\tilde C>0$, $\tilde C$
independent of $p$, and $\bar p>n$ such that
$$|Du_p(x)|\geq \tilde C\qquad\forall x\in\partial\Omega,
\quad\forall p\geq\bar p.\eqno(3)$$
\noindent
{\bf Proof.} There exists $R>0$ such that for any $x_0\in
\partial\Omega$ there exists a ball of center $y\in\Omega$
and radius $R$ such that $B_R(y)\subset\Omega$ and
$x_0\in\partial B_R(y)$. Let, for instance,
$x_0\in\partial\Omega_1$ and $y$ as above.
Let us define, for $p>n$,
$w(x)=K\left(R^{{p-n \over p-1}}-
r^{{p-n \over p-1}}\right)$, where $r=|x-y|$,
${R \over 2}__0$.
For sufficiently large $p$, $u_p\geq{\gamma \over 2}$
in $\partial B_{R \over 2}$ so that, choosing
$K={\gamma \over 2}\left(R^{{p-n \over p-1}}-
\left({R \over 2}\right)^{{p-n \over p-1}}\right)^{-1}$,
we have $w\leq u_p$ on $\partial B_{R \over 2}$.
Moreover, $w\equiv 0\leq u_p$ on $\partial B_R$
and $\Delta_p w=0$ in $A$. The comparison principle
for solutions to equation $(1_p)$ implies that $w\leq u_p$
in $A$. Since $w(x_0)=
u_p(x_0)=0$, we have
$${\partial u_p \over \partial {\bf n}}(x_0)\geq
-w_r(x_0)={\gamma \over 2}\left({p-n \over p-1}\right)
R^{1-n \over p-1}
\left(R^{p-n \over p-1}-{R \over 2}^{p-n \over p-1}\right)
^{-1}\rightarrow{\gamma \over R},$$
as $p\rightarrow\infty$, where ${\bf n}$
is the outer unit normal to $\Omega_1$, and (3) follows.
\proclaim
Theorem 1. Let $\Omega$ be as in the hypotheses of
Lemma 2 and moreover let us assume that
the starshapedness condition (S) holds.
Let $u$ be the solution to the Dirichlet problem
$(1_\infty)$-(BC) extended by $0$ to all of $\bar\Omega_2$.
Then there exist a constant $\delta>0$,
$\delta$ independent of $p$,
and $\bar p>n$ such that
$${\partial u_p \over \partial r}(x)\geq \delta
\qquad \forall x\in\Omega,\quad
\forall p\geq\bar p,\eqno(4)$$
$${\partial u \over \partial r}(x)\geq \delta
\qquad \hbox{ a.e. } x\in\Omega,\eqno(5)$$
where $u_p$ is the solution to $(1_p)$-(BC).
Moreover the set
$\Omega_k=\{x\in \Omega_2 \ |\ u(x) C'$ almost everywhere, for any $C'__