Infinitely many solutions for a class of semilinear elliptic equations

In this paper we find some new conditions to ensure the existence of infinitely many nontrivial solutions for the Dirichlet boundary value problems of the form −Δu+a(x)u=g(x,u)$-\mathrm{\Delta }u+a(x)u=g(x,u)$ in a bounded smooth domain. Conditions (_S1)$(S_1)$–(_S3)$(S_3)$ in the present paper are somewhat weaker than the famous Ambrosetti–Rabinowitz-type superquadratic condition. Here, we assume that the primitive of the nonlinearity g   is either asymptotically quadratic or superquadratic as |u|→∞$|u|\to \infty$.