Existence of groundstates for a class of nonlinear Choquard equations

Moroz, Vitaly; Van Schaftingen, Jean

doi:10.1090/S0002-9947-2014-06289-2

Existence of groundstates for a class of nonlinear Choquard equations
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Trans. Amer. Math. Soc. 367 (2015), 6557-6579 Request permission

Abstract:

We prove the existence of a nontrivial solution $u \in H^1 (\mathbb {R}^N)$ to the nonlinear Choquard equation \[ - \Delta u + u = \bigl (I_\alpha \ast F (u)\bigr ) F’ (u) \quad \text {in $\mathbb {R}^N$,} \] where $I_\alpha$ is a Riesz potential, under almost necessary conditions on the nonlinearity $F$ in the spirit of Berestycki and Lions. This solution is a groundstate and has additional local regularity properties; if moreover $F$ is even and monotone on $(0,\infty )$, then $u$ is of constant sign and radially symmetric.

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