Nodal solutions for the Choquard equation

We consider the general Choquard equations−Δu+u=(Iα⁎|u|p)|u|p−2u where Iα is a Riesz potential. We construct minimal action odd solutions for p∈(N+αN,N+αN−2) and minimal action nodal solutions for p∈(2,N+αN−2). We introduce a new minimax principle for least action nodal solutions and we develop new concentration-compactness lemmas for sign-changing Palais-Smale sequences. The nonlinear Schrödinger equation, which is the nonlocal counterpart of the Choquard equation, does not have such solutions.