Existence of weak solutions for variable exponent Schrödinger–Maxwell systems

This paper investigates the existence of energy solutions for a Schrödinger-Maxwell system with a variable exponent on a bounded domain D⊂ RN ( N≥ 2 ), where the p(x)-Laplacian couples the unknown functions u and v. Under the assumptions that f∈ Lm(· )(D) with $$m(x)>N/p(x)$$ and that the continuous exponent satisfies p(· )>1 , a variational approach is employed to establish the existence of weak solutions (u,v)∈ ( W₀1,p(· )(D)) ² for the considered system. Furthermore, it is shown that this solution is a saddle point of the associated energy functional.