Existence of solutions for a nonhomogeneous Dirichlet problem involving $p(x)$-Laplacian operator and indefinite weight

Abstract We obtain multiplicity and uniqueness results in the weak sense for the following nonhomogeneous quasilinear equation involving the $p(x)$ p ( x ) -Laplacian operator with Dirichlet boundary condition: $$ -\Delta _{p(x)}u+V(x) \vert u \vert ^{q(x)-2}u =f(x,u)\quad \text{in }\varOmega , u=0 \text{ on }\partial \varOmega , $$ − Δ p ( x ) u + V ( x ) | u | q ( x ) − 2 u = f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in $\mathbb{R}^{N}$ R N , V is a given function with an indefinite sign in a suitable variable exponent Lebesgue space, $f(x,t)$ f ( x , t ) is a Carathéodory function satisfying some growth conditions. Depending on the assumptions, the solutions set may consist of a bounded infinite sequence of solutions or a unique one. Our technique is based on a symmetric version of the mountain pass theorem.