0$, then (ii)\quad $u$ is symmetric and decreasing in $y$. Finally, in the case of an $N$-ball $(\Omega=B_{R})$ with a positive radius $R$ and with $f=f(|x|,u)$ and $F=\phi(r)f(|x|,u)$ monotonously nonincreasing in $|x|$, then (iii)\quad $u$ is locally symmetric in every direction. \endproclaim \noindent{\bf {Proof}}:\newline (i)\quad\quad Since $u$ is an element of ${W_{0}}^{1,p}(\Omega),$ then $u^{t}$ is an element of ${W_{0}}^{1,p}(\Omega)$ and by (4), we obtain, $$\int\limits_{\Omega} a(u, |\nabla\,u|) |\nabla\,(u^{t}-u)|\, d\text{\bf x} \geq \int\limits_{\Omega}{\ \phi(r) f(x,u)}(u^{t}-u)\, d\text{\bf x},$$ for all $t \in [0,+ \infty]$. Using the inequality $$\int\limits_{\Omega}{\ \phi(r) f(x,u)} \, d \text{\bf x} \leq \int\limits_{\Omega^{t}}{\ \phi(r) f(x,u^{t})}\, d \text{\bf x} ,$$ we conclude that $$\lim_{t \rightarrow 0} \frac{1}{t} \int\limits_{\Omega} a(u^{t}, |\nabla\,u^{t}|) |\nabla\,(u^{t})|^{2} \,d \text{\bf x} - \int\limits_{\Omega} a(u, |\nabla\,u|) |\nabla\,(u)|^{2} \, d\text{\bf x}=0.$$ Consequently, $u$ is locally symmetric, which is the desired result.\newline (ii)\quad\quad If $f$ is nondecreasing in the positive variable $y$, we can find $$x^{1}=(x_{0}',y_{1}),\,\, x^{2}=(x_{0}',y_{2})\,\text{in} \,\,\Omega,$$ with $$y_{1}+y_{2} \ne 0. \eqno(40)$$ By the hypothesis on $u$, $\frac{\partial u}{\partial y} > 0$ at $x^{1}$. \newline Let $U_{1}$ denote the (maximal) connected component of $\Omega \cap \left\{ x: u_{y}(x) > 0 \right\}$ containing $x^{1}$, where $x^{2}=(x_{0}',y_{2}) \in \Omega$, $y_{1} < y_{2}$,\newline and $u(x^{1})=u(x^{2}) < u(x_{0}',y)$ for all $y$ in $(y_{1},y_{2}).$ Then, for all $(x^{1},y)\in U_{1}$, $u(x^{1},y)=(x^{1},y_{1} + y_{2} - y) < u(x^{1},z)$ for all $z$ in $(y,y_{1}+ y_{2} - y).$ \newline We put $v(x^{1},y)=u(x^{1},y_{1}+ y_{2} - y)$ and apply Theorem 13 of Brock to conclude that $v(x)=u(x)$ for all $x \in U \subset V(x^{1})$. Using the equation in the statement of the problem, $w:=u-v$ which is zero. Then, $$-\text{Div} (a(w,{|{\nabla w}|}) {|\nabla w|})=\phi(r)[f(x^{1},y,u)-f(x^{1},y_{1} + y_{2} - y,u)]\quad\text{in}\quad U^{1},$$ which contradicts (40).\newline (iii)\quad\quad Let $\Omega$ be the ball $B_{R}$ and $f=f(|x|,u)$. If we associate to $x$ the value $\xi$ defined by $\xi:=({\xi}^{1},\eta)$ for an arbitrary rotation of the coordinate system about the origin, we see that f is not even and nonincreasing in $\eta$. By the above considerations, this yields the last assertion of the theorem. \vskip 0.3 cm {\bf Acknowledgement:} $\ $ I thank F. Brock from Leipzig, for helpful discussions. Also my particular thanks go to the referees for their remarks and suggestions. \vskip 0.5 cm \Refs \ref \no 1 \by F. Brock \pages 25--48 \paper Continuous Steiner symmetrization \yr 1995 \vol 172 \jour Math Nachr \endref \ref \no 2 \by F. Brock \paper Continuous symmetrization and symmetry of solutions of elliptic problems \yr 1998 \vol 124 \jour submitted to: Memoirs of A.M.S. \endref \ref \no 3 \by A. Henrot and G. A. Philippin \paper On a class of overdetermined eigenvalue problems \pages 905--914 \jour Mathematical Methods in the Applied Sciences \yr 1997 \vol 20(11) \endref \ref \no 4 \by A. Ladyzhenskaya and N. Ural' Tseva \book Linear and Quasilinear Elliptic Equations \publ Leningrad State University, Leningrad, U.S.S.R. \endref \ref \no 5 \by E. Mitidieri \paper A Rellich type identity and Applications \pages 125--151 \jour Commun. in Partial Differential \yr 1993 \vol 18(1 and 2) \endref \ref \no 6 \by F. Murat and J. Simon \paper Sur le contr\^ole par un domaine g\'eom\'etrique \jour Publication du Laboratoire d'Analyse Num\'erique de l'Universit\'e Paris 6 \yr 1976 \vol 189 \endref \ref \no 7 \by J. Serrin \paper A symmetry problem in potential theory \jour Arch. Rational Mech. Anal. \vol 43 \yr 1971 \pages 304--318 \endref \ref \no 8 \by J. Simon \paper Differentiation with respect to the domain in boundary value problems \jour Num. Funct. Anal. Optimz. \vol 2(7,8) \yr 1980 \pages 649--687 \endref \ref \no 9 \by J. Sokolowski and J. P. Zolesio \book Introduction to shape optimization: shape sensitivity analysis \publ Springer Series in Computational Mathematics, Vol. 10, Springer Berlin \yr 1992 \endref \endRefs \enddocument \bye