Abstract In this paper, we concern the existence result of the following general eigenvalue problem: { 𝒜 ( u ) = λ ℬ ( u ) in Ω , D α ( u ) = 0 on ∂ Ω , \left\{\begin{aligned} \displaystyle{}\mathcal{A}(u)&\displaystyle={\lambda}% \mathcal{B}(u)&&\displaystyle\phantom{}\text{in }{\Omega},\\ \displaystyle D^{\alpha}(u)&\displaystyle=0&&\displaystyle\phantom{}\text{on }% {\partial\Omega},\end{aligned}\right. in an arbitrary Musielak–Orlicz spaces, where 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} are quasilinear operators in divergence form of order 2 n {2n} and 2 ( n - 1 ) {2(n-1)} , respectively. The main assumptions in this case are that 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} are potential operators with 𝒜 {\mathcal{A}} being elliptic and monotone. In this study, we intentionally avoid imposing constraints on the growth of a generalized N-function, including the Δ 2 {\Delta_{2}} -condition for both the generalized N-function and its conjugate. Consequently, this necessitates the formulation of the approximation theorem and the extensive utilization of modular convergence concepts.
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