Abstract

We study the following nonlinear Choquard equation
−∆u + V u = (Iα ∗ |u|
p
)|u|
p−2u in Ω ⊂ R
N ,

where N ≥ 2, p ∈ (1, +∞) and V (x) is a continuous radial function such that infx∈Ω V > 0.

First, assuming to have Neumann or Dirichlet boundary conditions, we prove existence of a positive radial solution when Ω is a ring-shaped domain or an exterior domain of the form RN \ Br(0). We also provide a nonexistence result: if p ≥N+αN−2 the corresponding Dirichlet problem does not have any nontrivial regular solution in strictly star-shaped domains. Finally, when considering annular domains, letting α → 0+ we obtain an existence result for the corresponding local problem with power-type nonlinearity.

This talk is based on a joint work with A. Cesaroni.

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