Perturbation for a p(x)p(x)-Laplacian equation involving oscillating nonlinearities in RNRN

In this paper we study multiple solutions of the following problem in RNRN: equation(0.1){−div(|∇u|p(x)−2∇u)+|u|p(x)−2u=f(x,u)+λg(x,u),u∈W1,p(x)(RN), where the potential F(x,t)=∫0tf(x,s)ds has a suitable oscillating behavior in any neighborhood of the origin (respectively + ∞), and gg is a perturbation term. Our results are a generalization of the case of the constant exponent and bounded domain from [G. Anello, G. Cordaro, Perturbation from Dirichlet problem involving oscillating nonlinearities, J. Differential Equations 234 (2007) 80–90] to the case of variable exponent and RNRN.