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AbstractWe 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:  −Δp(x)u+V(x)|u|q(x)−2u=f(x,u)in Ω,u=0 on ∂Ω,$$ -\Delta _{p(x)}u+V(x) \vert u \vert ^{q(x)-2}u =f(x,u)\quad \text{in }\varOmega , u=0 \text{ on }\partial \varOmega , $$ where Ω is a smooth bounded domain in RN$\mathbb{R}^{N}$, 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.

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