Caccioppoli-type inequalities for Dirac operators

Abstract

In this paper, we establish the Caccioppoli estimates for the nonlinear differential equation $$ - \overline{D}\bigl( \vert Dv \vert ^{p-2}Dv\bigr) = \lambda \vert v \vert ^{p-2}v, \quad 1< p< \infty ,$$ − D ‾ ( | D v | p − 2 D v ) = λ | v | p − 2 v , 1 < p < ∞ , where D is the Dirac operator. Moreover, we obtain general weighted versions of the Caccioppoli-type inequalities for the Dirac operators.