A class of eigenvalue problems for $$(\phi ,\overrightarrow{\tau }(\cdot ))$$-fractional spaces

Abstract In this paper, we are concerned with a new class of $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> -fractional spaces involving anisotropic $$\overrightarrow{\tau }(\cdot )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -Laplacian operators, abbreviated as ( $$\phi ,\overrightarrow{\tau }(\cdot )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> )-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 $$\tau _{M}^{+}&lt;q^{-}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>τ</mml:mi> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo>&lt;</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mo>-</mml:mo> </mml:msup> </mml:mrow> </mml:math> . Our main results are novel and contribute to the literature on problems involving ( $$\phi ,\overrightarrow{\tau }(\cdot )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> )-HFDA. This investigation enhances the understanding of this specific class of problems.