# Some elliptic problems with singular nonlinearity and advection for Riemannian manifolds

## TL;DR

This paper investigates the regularity, critical dimensions, and solution properties of singular semilinear elliptic problems with advection on Riemannian manifolds, providing new estimates and multiplicity results.

## Contribution

It establishes uniform Lebesgue estimates, identifies critical dimensions, and analyzes solution symmetry and multiplicity for these elliptic problems on Riemannian manifolds.

## Key findings

- Extends regularity results to Riemannian manifolds with zero Dirichlet boundary conditions.
- Determines critical dimensions for Gelfand, MEMS, and power nonlinearities.
- Proves multiplicity and uniqueness of solutions near critical parameters.

## Abstract

We are interested in regularity properties of semi-stable solutions for a class of singular semilinear elliptic problems with advection term defined on a smooth bounded domain of a complete Riemannian manifold with zero Dirichlet boundary condition. We prove uniform Lebesgue estimates and we determine the critical dimensions for these problems with nonlinearities of the type Gelfand, MEMS and power case. As an application, we show that extremal solutions are classical whenever the dimension of the manifold is below the critical dimension of the associated problem. Moreover, we analyze the branch of minimal solutions and we prove multiplicity results when the parameter is close to critical threshold and we obtain uniqueness on it. Furthermore, for the case of Riemannian models we study properties of radial symmetry and monotonicity for semi-stable solutions.