Multiple Solutions for Schrödinger Equations Involving Concave-Convex Nonlinearities Without (AR)-Type Condition

We obtain the multiplicity of solutions for the following Schr $$\ddot{\text{ o }}$$ dinger equations $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+V(x)u=g(x,u)&{}\text{ for } x\in {\mathbb {R}}^{N},\\ u(x)\rightarrow 0&{}\text{ as } |u|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where $$V\in C({\mathbb {R}}^{N},{\mathbb {R}})$$ is coercive at infinity and g involves concave–convex nonlinearities while the convex terms need not to satisfy the (AR)-type condition. Some new nonlinearities are considered and an example is given.