The objective of this chapter is to develop an approach for proving the existence of solutions to problems where the monotonicity property of the driving operator is absent. We focus on nonlinear boundary value problems driven by continuous, coercive operators that lack sufficient monotonicity or pseudo-monotonicity to permit the application of the Minty–Browder theorem, in either version discussed in the preceding chapter. Assuming that the given operator is continuous on finite-dimensional subspaces and coercive, we immediately obtain the solvability of each Galerkin-type approximation, along with the boundedness of the sequence of approximate solutions. Although we can extract a weakly convergent subsequence, we cannot assert the solvability of the original nonlinear problem in the usual (weak) sense. The concept of a solution, referred to as a generalized solution, is introduced in this chapter. We begin by describing competing operators and then consider a model boundary value problem driven by the competing (p, q)-Laplacian with Dirichlet boundary conditions, examining the convergence of the associated Galerkin scheme. Next, we introduce the notion of an abstract generalized solution, whose existence is guaranteed by the coercivity and continuity of the operator. In addition to the model problem, which also involves an unbounded weight, we present applications to competing fractional operators and mixed local–nonlocal competing problems.