Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth

where ω > 0 is a constant, λ > 0 is a positive parameter. Klein-Gordon equation can be used to develop the theory of electrically charged fields (see[13]) and study the interaction with an assigned electromagnetic field (see [11]). The Born-Infeld (BI) electromagnetic theory [2, 3] was originally proposed as a nonlinear correction of the Maxwell theory in order to overcome the problem of infiniteness in the classical electrodynamics of point particles(see [14]). Klein-Gordon equation

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