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

Abstract

The article deals with the existence of a pair of nontrivial nonnegative and nonpositive solutions for a nonlinear weighted quasilinear equation in RN{{\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)−2ξ{| \xi | }^{q\left(x)-2}\xi if ∣ξ∣| \xi | is small and like ∣ξ∣p(x)−2ξ{| \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.

Keywords

• double-phase
• variable exponent sobolev spaces
• critical points
• cerami condition
• primary 35j62
• 35j70
• secondary 35j20