AbstractWe consider the reaction-diffusion problem -Δgu = f(u) in ℬR with zero
Dirichlet boundary condition, posed in a geodesic ball ℬR with radius
R of a Riemannian model (M,g). This class of Riemannian manifolds includes the
classical space forms, i.e., the Euclidean, elliptic, and hyperbolic spaces.
For the class of semistable solutions we prove radial symmetry and monotonicity.
Furthermore, we establish L∞, Lp, and W1,p estimates which are optimal and do not
depend on the nonlinearity f. As an application, under standard assumptions on the nonlinearity
λf(u), we prove that the corresponding extremal solution u* is bounded whenever
n ≤ 9. To establish the optimality of our regularity results we find the extremal solution
for some exponential and power nonlinearities using an improved weighted Hardy inequality.
Some regularity results for p-harmonic mappings between Riemannian manifolds
