On some multiple solutions for a p(x)-Laplacian equation with critical growth

In this work we prove that, for each m∈Nm∈N, if λ is small, there are m   solutions for the equation −div(|∇u|p(x)−2∇u)=f(x,u)+λ|u|q(x)−2u−div(|∇u|p(x)−2∇u)=f(x,u)+λ|u|q(x)−2u, x∈Ωx∈Ω, in a bounded domain of RNRN, the nonlinearity is given by the critical growth in the context of variable exponents p⁎(x)=Np(x)/(N−p(x))p⁎(x)=Np(x)/(N−p(x)). The main tools used are the variational method and concentration compactness principle.