Nodal solutions for the Choquard equation

Abstract We consider the general Choquard equations − Δ u + u = ( I α ⁎ | u | p ) | u | p − 2 u where I α is a Riesz potential. We construct minimal action odd solutions for p ∈ ( N + α N , N + α N − 2 ) and minimal action nodal solutions for p ∈ ( 2 , N + α N − 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 Schrodinger equation, which is the nonlocal counterpart of the Choquard equation, does not have such solutions.

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