Logarithmic double phase problems with convection: existence and uniqueness results

In this paper, we deal with a problem driven by a logarithmic double phase operator with variable exponent and a nonlinearity on the right-hand side which also depends on the gradient of the solution. Under very general structure conditions, following a topological approach based on the use of pseudomonotone operators, we establish the existence of a nontrivial weak solutions for the problem under consideration. Moreover, imposing more restrictive conditions on the nonlinearity, we are able to provide the uniqueness of the solution. Finally, we prove the boundedness, closedness and compactness of the related solution set to our problem.