Nonexistence results for a nonlocal weighted quasilinear elliptic differential inequality with gradient absorption terms in bounded domain

This work is devoted to the nonexistence of positive weak solutions for a weighted quasilinear elliptic differential inequality involving nonlocal source and gradient absorption terms in bounded domain, where positive weight functions may be singular or degenerate at the boundary. Under the condition that the weight function of the absorption term is either a sufficiently small positive constant or a more general assumption, we establish new nonexistence results containing the critical cases. The key ingredient in the proof is the rescaled test function method developed by Mitidieri and Pohozaev, which is essentially based on nonlinear capacity estimates adapted to the nonlocal setting of our bounded domain problem.