Elliptic problems with unbalanced growth, unbounded coefficient and intrinsic operator

We investigate a variable exponent double phase problem involving an unbounded coefficient in the principal part and a multi-valued convection term defined via an intrinsic operator. Under suitable growth and coercivity conditions, truncation techniques and a surjectivity result for multivalued pseudomonotone operators yield bounded weak solutions with bounded boundary traces. Furthermore, we examine special cases of the double phase problem by selecting distinct intrinsic operators, including convolution-type operators. The analysis is motivated by [Math. Z. 308 (2024), no. 2, Paper No. 38, 13 pp.] and incorporates mixed boundary conditions and multivalued source terms.