Mixed local-nonlocal eigenvalue problems in the Heisenberg group: spectral theory and singular multiplicity

RESUMO

This paper investigates nonlinear eigenvalue problems governed by a family of operators that combine two distinct diffusion mechanisms: a classical local diffusion, which accounts for short-range interactions, and a fractional nonlocal diffusion, which captures long-range effects. The analysis is carried out in the Heisenberg group, a non-Euclidean geometric setting in which motion is constrained to a distinguished set of horizontal directions. As an application of the obtained spectral theory, we establish multiplicity results for perturbed singular problems in both the purely nonlocal and mixed local--nonlocal settings. For the mixed case, a gradient convergence theorem plays a crucial role in deriving these multiplicity results. A key feature of our analysis is the treatment of variable singular exponents, allowing the singularity to vary throughout the domain. To the best of our knowledge, this is the first systematic study of nonlinear eigenvalue problems and singular problems with variable singularity exponents for mixed local-nonlocal operators in the Heisenberg group. The results therefore open a new direction in the spectral theory of subelliptic operators and provide a rigorous mathematical foundation for applications ranging from anomalous diffusion and nonlocal phase transitions to image analysis and control theory on nonholonomic systems.