A $$\boldsymbol{\Im}$$-Hilfer Generalized Kirchhoff-Double Phase Problems with $$\boldsymbol{\mathfrak{p}}$$-Laplacian Operator

The aim of this work is to investigate the existence and multiplicity of weak solutions for a $$\Im$$ -Hilfer generalized Kirchhoff-double phase problem. Under suitable hypotheses, we employ the sub-supersolution method combined with monotone operator theory, the maximum principle, and minimization arguments to establish the existence of at least one nontrivial solution. With additional assumptions, we prove the existence of a second solution using the mountain pass theorem. To the best of our knowledge, these results are novel for Kirchhoff-double phase problems involving variable exponents, contributing significantly to the existing literature.