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If a Dirichlet problem (elliptic PDE, in $R^{n}$) is solved by transforming into ODE (proving its radial symmetry) how can we study it on manifold?

For example, $B$ is the unit ball in $R^{n}$, the Gelfand problem is

$$ \left\{\begin{aligned} -\Delta u & =\lambda e^u & & \text { in } B, \\ u & =0 & & \text { on } \partial B, \end{aligned}\right. $$

it can be transformed into

$$ \left\{\begin{array}{c} r^{-(N-1)}\left(r^{N-1} u^{\prime}\right)^{\prime}+\lambda e^u=0 \\ u(0)=a \quad u^{\prime}(0)=0, \end{array}\right. $$

if we have proved its radial symmetry, then we can study this ODE to get some results for the original PDE, I wonder if this process can be applied on the manifold, for example, compact riemannian manifold M without boundary $$ -\Delta u=\lambda e^u \quad \text { on } M. $$

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    $\begingroup$ The answer will depend entirely on whether the Riemannian manifold $(M,g)$ itself has spherical/rotational symmetry ("radial symmetry" as you wrote). $\endgroup$
    – Igor Khavkine
    Commented Dec 6, 2022 at 11:39
  • $\begingroup$ I‘m a little confused about the definition of the spherically symmetry Riemannian manifold, in math.stackexchange.com/questions/4044861/… there are some useful discussions. $\endgroup$
    – Elio Li
    Commented Dec 9, 2022 at 4:20
  • $\begingroup$ If the Riemannian manifold does not have spherical symmetry, it's unlikely that harmonic functions will have that symmetry. Spherical symmetry is simply an action of the orthogonal group by isometries. $\endgroup$
    – Igor Khavkine
    Commented Dec 9, 2022 at 10:41

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