Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition

The article deals with the existence of a pair of nontrivial nonnegative and nonpositive solutions for a nonlinear weighted quasilinear equation in RN, which involves a double-phase general variable exponent elliptic operator A. More precisely, A has behaviors like |xi| (q(x )-2) xi if |xi | is small and like |xi| (p(x )-2) xi if |xi | is large. Existence is proved by the Cerami condition instead of the classical Palais-Smale condition, so that the nonlinear term f(x, u) does not necessarily have to satisfy the Ambrosetti-Rabinowitz condition.