Schrödinger equations with concave and convex nonlinearities

Abstract.

We obtain the existence of infinitely many nodal solutions for the Schrödinger type equation on $\mathbb{R}^N : - \Delta u + V(x)u = f(x,u)$ with $u \in H^1 (\mathbb{R}^N ).$ Here, $V \in C(\mathbb{R}^N ,\mathbb{R}),\,V(x) \geq 1,\int_{\mathbb{R}^N } {(V(x))^{ - 1} dx < + \infty .}$ The nonlinearity f is symmetric in the sense of being odd in u, and may involve a combination of concave and convex terms.