AProof of Gidas-Ni-Nirenberg theorem
Suppose $u$ is a smooth, positive solution in the unit ball $B_1$ to the nonliner elliptic equation
and $u = 0$ on $\partial B_1$. If $f$ is Lipschitz, then $u$ is radially symmetric.
在 Bath 大学的一门课程 Theory of Partial Differential Equation中, 有一个章专门讲这个的.
Necessity of $u>0$ in Theorem. Give an explicit counterexample showing that the conclusion of the GNN theorem can fail when the assumption that $u>0$ in $B$ is dropped.
Let $u = sinx$, $\Delta u = -sinx = -u$, but $u$ is not even. It shows that $u$ is not radially symmetric.
