Coupled Klein–Gordon and Born–Infeld type equations: looking for solitary waves

where u is a real function and ω ∈ R. If one looks for solutions of (1.1) having the form (1.2), the nonlinear Klein-Gordon equation reduces to a semilinear elliptic equation, as well as if one looks for solitary waves of nonlinear Schrödinger equation (see [10], [12] and the papers quoted therein). Many existence results have been established for solutions u of such a semilinear equation, both in the case in which u is radially symmetric and real or non-radially symmetric and complex (e.g., see [6], [7], [15]). From equation (1.1) it is possible to develop the theory of electrically charged fields (see [13]) and study the interaction of ψ with an assigned electromagnetic field (see [1], [2], [9]). On the other hand, it is also possible to study the interaction of ψ with its own electromagnetic field (see [3], [4], [5], [10]), which is not assigned, but is an unknown of the problem. More precisely, if the electromagnetic field is described by the gauge potentials (φ,A)