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This paper is divided in two parts. In the first part, we prove coercivityresults and minimization of the Euler energy functional. In the second part, wefocus on the existence and multiplicity of a positive solution of fractionalDirichlet problem involving the $\gamma(\xi)$-Laplacian equation withnon-negative weight functions in$\mathcal{H}^{\alpha,\beta;\chi}_{\gamma(\xi)}(\Lambda,\mathbb{R})$ using somevariational techniques and Nehari manifold.

Existence and multiplicity for fractional Dirichlet problem with $\gamma(\xi)$-Laplacian equation and Nehari manifold