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

Abstract

Abstract The article deals with the existence of a pair of nontrivial nonnegative and nonpositive solutions for a nonlinear weighted quasilinear equation in R N {{\mathbb{R}}}^{N} , which involves a double-phase general variable exponent elliptic operator A {\bf{A}} . More precisely, A {\bf{A}} has behaviors like ∣ ξ ∣ q ( x ) − 2 ξ {| \xi | }^{q\left(x)-2}\xi if ∣ ξ ∣ | \xi | is small and like ∣ ξ ∣ p ( x ) − 2 ξ {| \xi | }^{p\left(x)-2}\xi if ∣ ξ ∣ | \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 ) f\left(x,u) does not necessarily have to satisfy the Ambrosetti-Rabinowitz condition.