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This paper is devoted to studying the existence and multiplicity of nontrivial solutions for the following boundary value problem cases -div(ω(x)| abla u(x)|^N-2 abla u(x))=f(x,u)+ε h(x), & in B; u=0, & on B, caseswhere B is the unit ball in R^N, the radial positive weight ω(x) is of logarithmic type function, the functional f(x,u) is continuous in B×R and has double exponential critical growth, which behaves like exp{e^α|u|^NN-1 } as |u| → ∞ for some α > 0. Moreover, ϵ>0, and the radial function h belongs to the dual space of W^1,N_0,rad(B) h e 0. , number=1, journal=Journal of Partial Differential Equations, year=2025, month=Apr., pages=80–100