Existence and multiplicity of solutions for a class of quasilinear elliptic equations: An Orlicz–Sobolev space setting

In this paper, we study the existence and multiplicity of solutions of the following quasilinear elliptic problem { − div ( a ( | ∇ u | ) ∇ u ) = f ( x , u ) , in Ω , u = 0 , on ∂ Ω , where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω. The existence and multiplicity of solutions are obtained by genus theory, symmetric mountain pass lemma, fountain theorem and dual fountain theorem, respectively.