TY  - JOUR
ID  - Garain2026
LB  - Garain2026
AU  - Garain, Prashanta
TI  - On mixed local-nonlocal Sobolev-type inequalities and their connection with singular equations in the Heisenberg group
ST  - On mixed local-nonlocal Sobolev-type inequalities and their connection with singular equations in the Heisenberg group
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2026
VL  - 46
IS  - 3
SP  - 305
EP  - 321
DO  - https://doi.org/10.7494/OpMath.202603271
UR  - https://doi.org/10.7494/OpMath.202603271
AB  - In this work, we establish a mixed local-nonlocal Sobolev-type inequality in the Heisenberg group and demonstrate that its extremals coincide with solutions to the corresponding mixed local-nonlocal singular \(p\)-Laplace equations. We further show that these inequalities serve as a necessary and sufficient condition for the existence of weak solutions to the associated singular problems. Notably, the same characterization remains valid in both the purely local and purely nonlocal settings. Our results thus provide a unified framework linking the existence theory for singular equations across local, nonlocal, and mixed regimes.
KW  - Sobolev type inequality
KW  - extremal
KW  - mixed local-nonlocal singular problem
KW  - Heisenberg group
SN  - 1232-9274
ER  -