Quasilinear problems with superlinear or asymptotically linear nonlinearities

We establish the existence of a nontrivial solution for the quasilinear problem [Formula: see text] where [Formula: see text] is a smooth bounded domain, [Formula: see text] is a generalized Orlicz function, and [Formula: see text] denotes the associated [Formula: see text]-Laplacian operator. The nonlinearity [Formula: see text] is assumed to be either superlinear or asymptotically linear at both infinity and the origin. We prove the existence of bound and ground state solutions. The main difficulty here comes from the fact [Formula: see text] can be a generalized Orlicz function which covers the double phase problem.