Existence of solutions for p(x)-Laplacian equations with singular coefficients in RN

In this paper, we deal with the existence of solutions for the following p(x)-Laplacian equations via critical point theory{−div(|∇u|p(x)−2∇u)+e(x)|u|p(x)−2u=f(x,u)in RN,u∈W1,p(x)(RN), where f(x,u)=∑i=1mλiai(x)gi(x,u), gi:RN×R→R satisfies the Caratheodory condition, but ai(x) are singular. Especially, we obtain existence criterion for infinite many pairs of solutions for the problem, when some ai0(x) can change sign and gi0(x,⋅) satisfies super-p+ growth condition.