Coupled Klein–Gordon and Born–Infeld type equations: looking for solitary waves By Dimitri Mugnai Dipartimento di Matematica e Informatica Universit`a di Perugia, Via Vanvitelli 1, Perugia 06123 (Italy), e-mail: mugnai@dipmat.unipg.it The existence of infinitely many nontrivial radially symmetric solitary waves for the nonlinear Klein-Gordon equation, coupled with a Born-Infeld type equation, is established under general assumptions. Keywords: Z 2 –Mountain Pass, L ∞ –a priori estimate. 1. Introduction Let us consider the following nonlinear Klein-Gordon equation: ∂ 2 ψ ∂t 2 - Δψ + m 2 ψ -|ψ| p-2 ψ =0, (1.1) where ψ = ψ(t, x) ∈ C, t ∈ R, x ∈ R 3 , m ∈ R and 2 <p< 6. In the last years a wide interest was born about solitary waves of (1.1), i.e. solutions of the form ψ(x, t)= u(x)e iωt , (1.2) 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 Schr¨odinger 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) φ : R 3 × R -→ R, A : R 3 × R -→ R 3 , then, by Maxwell equations, the electric field is given by E = -∇φ - ∂ A ∂t Article submitted to Royal Society T E X Paper