The main crux of this manuscript is to study the existence of weak solutions for the following fractional double phase elliptic problem with Dirichlet boundary conditions \(\begin{aligned} {\left\{ \begin{array}{ll}(-\Delta )_{p}^\sigma w + (-\Delta )_{q}^\sigma w = \psi (z) + g(z,w) & \text{ in } \Theta , \\ w=0, & \text{ on } \mathbb {R}^N \backslash \Theta ,\end{array}\right. } \end{aligned}\) where \(1<p<q < \frac{N}{\sigma }\) , \(\sigma \in (0,1)\) and \((-\Delta )_{p}^\sigma , (-\Delta )_{q}^\sigma \) are fractional Laplacian operators defined in the principle value sense. The existence results are obtained by combining the Galerkin approximation, Lebesgue and Sobolev spaces, the Young measures approach, and the assumption of growth conditions on a given datum \(\varvec{g}\) . Our results expand upon and broaden several recent studies in this area of literature.
