Infinite solutions having a prescribed number of nodes for a Schrödinger problem

In this paper, we are concerned with the multiplicity of solutions for a Schrodinger problem. A weaker super-quadratic assumption is required for the nonlinearity. Then we give a new proof for the infinite solutions to the problem, having a prescribed number of nodes. It turns out that the weaker condition of the nonlinearity suffices to guarantee the infinitely many solutions. At the same time, a global characterization of the critical values of the non-radial nodal solutions are given.