A group-theoretical approach for nonlinear Schrödinger equations

The purpose of this paper is to study the existence of weak solutions for some classes of Schrödinger equations defined on the Euclidean space {\mathbb{R}^{d}} ({d\geq 3}). These equations have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using the Palais principle of symmetric criticality and a group-theoretical approach used on a suitable closed subgroup of the orthogonal group {O(d)}. In addition, if the nonlinear term is odd, and {d>3}, the existence of {(-1)^{d}+[\frac{d-3}{2}]} pairs of sign-changing solutions has been proved. To make the nonlinear setting work, a certain summability of the {L^{\infty}}-positive and radially symmetric potential term W governing the Schrödinger equations is requested. A concrete example of an application is pointed out. Finally, we emphasize that the method adopted here should be applied for a wider class of energies largely studied in the current literature also in non-Euclidean setting as, for instance, concave-convex nonlinearities on Cartan–Hadamard manifolds with poles.