Nonvariational double phase problems with variable exponents depending on the gradient of the solution with convection term

In this work, we study nonvariational double phase problems driven by a novel operator with variable exponents depending on the solutions and its gradient, along with a convection term. Using the Galerkin method, we prove the existence of solutions and apply a truncation technique to demonstrate multiplicity. Additionally, we address the uniqueness of solutions. The paper presents self-contained techniques to prove the multiplicity result, which may also be useful for addressing other problems with a convection term.