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) = m i =1 λiai (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