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In this paper, we devote ourselves to considering a modified zero energy critical point theory for a specific set of functionals denoted as $\Phi_{\mu}$, defined within the confines of a uniformly convex Banach space. Integrating the nonlinear generalized Rayleigh quotient approach with Ljusternik–Schnirelman category, we establish the nonexistence and multiplicity of zero energy critical points of the involved functionals. In particular, the modified zero energy critical point theory can be applied to more nonlocal problems. Our main results improve and complement the existing results in the related literature.