  In this work we deal with elliptic equations driven by the variable exponent
double phase operator with a Kirchhoff term and a right-hand side that is just
locally defined in terms of very mild assumptions. Based on an abstract
critical point result of Kajikiya (2005) and recent a priori bounds for
generalized double phase problems by the authors (2022), we prove the existence
of a sequence of nontrivial solutions whose $L^\infty$-norms converge to zero.
