## Existence of groundstates for a class of nonlinear Choquard equations.(English)Zbl 1325.35052

Summary: 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.

### MSC:

 35J61 Semilinear elliptic equations 35B33 Critical exponents in context of PDEs 35B38 Critical points of functionals in context of PDEs (e.g., energy functionals) 35B65 Smoothness and regularity of solutions to PDEs 35Q55 NLS equations (nonlinear Schrödinger equations) 45K05 Integro-partial differential equations
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