6 papers
Survey on topological methods for Allen--Cahn equations and systems
João Henrique Andrade, Stefano Nardulli, Raoní Ponciano
We present a survey on multiplicity results for the Allen--Cahn equation and systems in the singular perturbation regime, emphasizing their geometric interpretation through -con…
Radial Sobolev embeddings on spherically symmetric Riemannian manifolds
João Marcos do Ó, Guozhen Lu, Raoní Ponciano
We study Sobolev spaces of radial functions on spherically symmetric Riemannian manifolds. Using geodesic polar coordinates, we give a sharp one-dimensional reduction: a radial fun…
Trudinger-Moser type inequalities for the Hessian equation with logarithmic weights
João Marcos do Ó, José Francisco de Oliveira, Raoní Cabral Ponciano
We establish sharp Trudinger-Moser inequalities with logarithmic weights for the -Hessian equation and investigate the existence of maximizers. Our analysis extends the classica…
From bubbles to clusters: Multiple solutions to the Allen--Cahn system
João Henrique de Andrade, Dario Corona, Stefano Nardulli +2
We extend previous works on the multiplicity of solutions to the Allen-Cahn system on closed Riemannian manifolds by considering an arbitrary number of phases. Specifically, we sho…
Sharp Sobolev and Adams-Trudinger-Moser inequalities for symmetric functions without boundary conditions on hyperbolic spaces
João Marcos do Ó, Guozhen Lu, Raoní Ponciano
Embedding theorems for symmetric functions without zero boundary condition have been studied on flat Riemannian manifolds, such as the Euclidean space. However, these theorems have…
Sharp higher order Admas' inequality with exact growth condition on weighted Sobolev spaces
João Marcos do Ó, Guozhen Lu, Raoní Ponciano
This paper introduces a novel higher order Adams inequality that incorporates an exact growth condition for a class of weighted Sobolev spaces. Our rigorous proof confirms the vali…