Critical concave-convex problems in Carnot groups

2025/12/04 by Galeotti, Mattia, Vecchi, Eugenio
#35B33 #35H20 #35J70 #35R03 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.04640

Abstract

We consider a model Dirichlet problem with concave-convex and critical nonlinearity settled in Carnot groups. Our aim is to prove the existence of two positve solutions in the spirit of a famous result by Ambrosetti, Brezis and Cerami. To this aim we use a variational Perron method combined with proper estimates of a family of functions which are minimizers of the relevant Sobolev inequality. Due to the lack of boundary regularity, we also have to be careful while proving that the first solution found is a local minimizer in the proper topology.

Citations

• Critical singular problems in Carnot groups
• Boundary regularity for subelliptic equations in the Heisenberg group
• On the existence of a second positive solution to mixed local-nonlocal concave-convex critical problems
• Struwe's Global Compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
• Higher order Boundary Schauder Estimates in Carnot Groups
• Green's identities, comparison principle and uniqueness of positive solutions for nonlinear p-sub-Laplacian equations on stratified Lie groups
• A nonlocal concave-convex problem with nonlocal mixed boundary data
• Schauder estimates at the boundary for sub-laplacians in Carnot groups
• Some existence results for the Webster scalar curvature problem in presence of symmetry

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• Critical singular problems in Carnot groups

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