Abstract In this paper, we are concerned with a new class of $$\phi $$ ϕ -fractional spaces involving anisotropic $$\overrightarrow{\tau }(\cdot )$$ τ → ( · ) -Laplacian operators, abbreviated as ( $$\phi ,\overrightarrow{\tau }(\cdot )$$ ϕ , τ → ( · ) )-HFDA, for differential equations with a power-like variable reaction term. By utilizing the Mountain Pass Theorem together with Ekeland’s Principle, we proof the existence of precise intervals of positive parameters that admit nontrivial solutions for an eigenvalue problem since $$\tau _{M}^{+}<q^{-}$$ τ M + < q - . Our main results are novel and contribute to the literature on problems involving ( $$\phi ,\overrightarrow{\tau }(\cdot )$$ ϕ , τ → ( · ) )-HFDA. This investigation enhances the understanding of this specific class of problems.