Nodal solutions for the Choquard equation

By Marco Ghimenti, Jean Van Schaftingen. 2015

228 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5904c8ee83f55f1648cb1152f03d88bf23077173

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Ground states and geometrically distinct solutions for periodic Choquard-Pekar equations (2020, background): “For the Choquard-type equation and related problems, we refer to [11,31,33,44] for the existence of solutions and multiplicity properties, to [13,22,47] for existence of sign-changing solutions, and to [12,32,45,48] for semiclassical solutions.”
Odd symmetry of least energy nodal solutions for the Choquard equation (2016, background): “Furthermore, there are also nodal solutions which minimize the energy in the so-called Nehari nodal set in the case N−2 N+α < 1 p ≤ 1 2 (see [10, 11]), which will be called least energy nodal solutions.”