On the Klein‐Gordon‐Maxwell system with critical exponential growth in ℝ2

Abstract

This paper is concerned with the following Klein‐Gordon‐Maxwell system: where ω > 0 is a constant, , and obeys exponential critical growth in the sense of the Trudinger‐Moser inequality. We give some new sufficient conditions on f, specifically related to exponential growth, to obtain the existence of nontrivial solutions. Furthermore, we prove a nonexistence result with a Pohozaev‐type argument which also provides important information to get the above existence results. In particular, some new analytical approaches and estimates are used to overcome the difficulty arising from the critical growth of Trudinger‐Moser type.