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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.

We consider a double-phase non-Newtonian fluid, described by a stress tensor which is the sum of a p-Stokes and a q-Stokes stress tensor, with 1 2 to justify calculations for weak solutions. Results are obtained through a careful use of the symmetries of the convective term and are also valid for rather general (even anisotropic) stress-tensors.