| Summary: | Abstract In this paper, we study the existence of infinitely many nontrivial solutions for the following semilinear Schrödinger equation: { − Δ u + V ( x ) u = f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , $$ \textstyle\begin{cases} -\Delta u+V(x)u=f(x,u), \quad x\in\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}), \end{cases} $$ where the potential V is continuous and is allowed to be sign-changing. By using a variant fountain theorem, we obtain the existence of infinitely many high energy solutions under the condition that the nonlinearity f ( x , u ) $f(x,u)$ is of super-linear growth at infinity. The super-quadratic growth condition imposed on F ( x , u ) = ∫ 0 u f ( x , t ) d t $F(x,u)=\int _{0}^{u}f(x,t)\,dt$ is weaker than the Ambrosetti–Rabinowitz type condition and the similar conditions employed in the references.
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