Solvability for Couple Nonlinear Elliptic Partial Differential Equations

As it famous the solvability of any mathematical problem play an important rule for many investigators whose interested about the numerical solution for such mathematical problems because it gives them the green light about the ability to solve such problems numerically, besides to give the green light for solving numerically many real life problems in various field of sciences; like the physics, medicine, astronomy, chemistry, biology, different branch of engineering and many other fields. For this reason, this work is devoted to studying the solvability (in infinite dimension) of a couple of nonlinear elliptic partial differential equations (CNLEPDEs) with four different types of boundary conditions (BCs): Dirichlet (DBCs), Neumann (NBCs), Robin (RBCs), and Mixed BCs (MBCs). The weak formulation (WF) for each problem is determined by the type of each B. The existence and the uniqueness of the solution for the proposed problem (CNLEPDEs) with each type of BCs is proved by employing the Minty-Browder theorem (MBTH), through using the Poincaré-Friedrichs inequality of (PFI) in the case of homogeneous BCs (HBCs) for each type, and through using the generalization of PFI (GPFI) and the trace Operator (TRO) in the case of nonhomogeneous BCs(NHBCs) for each type.