Existence of multiple weak solutions to the elliptic equations with nonstandard growth by the Fountain theorem

Abstract

We consider an elliptic problem

−div (a (x, ∇u)) = ηu|u|p(x)−2 + λu|u|z(x)−2 + cu|u|p (x)−2 ,
u|∂Ω = 0
in the domain Ω ⊂ Rl , l ≥ 3, under rather general conditions on the elliptic operator involving variable exponent conditions. We show the existence of a weak solution by means of the mountain pass theorem. Our main result is the establishment of the existence of infinitely many nontrivial weak solutions under the assumption of symmetry A(x, ξ) = A(x, −ξ) for all x ∈ Ω and all ξ ∈ Rl .