On superlinear p(x)p(x)-Laplacian equations in RN

We consider the p(x)p(x)-Laplacian equations in RN. The nonlinearity is superlinear but does not satisfy the Ambrosetti–Rabinowitz type condition. We obtain ground states of the equations, improving a recent result of Fan [X.L. Fan, p(x)p(x)-Laplacian equations in RN with periodic data and nonperiodic perturbations, J. Math. Anal. Appl. 341 (2008) 103–119]. We also establish a Bartsch–Wang type compact embedding theorem for variable exponent spaces. Then, a multiplicity result for the equations is proved for odd nonlinearity.