This paper deals with a \(p(\cdot ,\cdot )\) -fractional type equations involving logarithmic and critical nonlinearities on the entire space \(\mathbb {R}^{N}\) . The approach used combines the concentration–compactness principle with Mountain Pass Theorem to ensure the existence of a non-trivial ground state solution to the equation with concave–convex nonlinearities. Further, a basic example is presented to demonstrate the validity of our main theorem’s conditions.
