Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Regularity of stable solutions to semilinear elliptic equations on
  Riemannian models   We consider the reaction-diffusion problem $-\Delta_g u = f(u)$ in
$\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic
ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class
of Riemannian manifolds includes the classical \textit{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
$L^\infty$, $L^p$, and $W^{1,p}$ estimates which are optimal and do not depend
on the nonlinearity $f$. As an application, under standard assumptions on the
nonlinearity $\lambda f(u)$, we prove that the corresponding extremal solution
$u^*$ is bounded whenever $n\leq9$. 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.

22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
17
The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat
1
The categories of this paper are: cs
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin