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