Existence and \(L^{\infty }\)-Estimates of the Solutions to a Class of Inclusion Problems

We introduce a class of inclusion problems with the Dirichlet boundary involving nonlinear elliptic differential operators and multivalued convection terms in the generalized Orlicz–Sobolev spaces. Some sufficient conditions are provided to ensure that sub and supersolutions exist. We established existence theorems and estimated \(L^{\infty }\) $L^{\infty }$ norm of solutions to the inclusion problems via the method of sub-supersolution. Some examples of applications for the above inclusion problems are presented.