The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation. (English) Zbl 1397.35087
Summary: We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation
\[
- \Delta u = \left( \int_\Omega \frac{| u(y)|^{2_\mu^*}}{| {x - y} |^\mu}dy \right)| u|^{2_\mu^* - 2}u + \lambda u \quad\text{in }\Omega,
\]
where \(\Omega\) is a bounded domain of \(\mathbb R^N\) with Lipschitz boundary, \(\lambda\) is a real parameter, \(N\geqslant 3\), \(2_\mu^* = \left( {2N - \mu} \right)/\left( {N - 2} \right)\) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.
MSC:
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
Keywords:
Brezis-Nirenberg problem; Choquard equation; Hardy-Littlewood-Sobolev inequality; critical exponentReferences:
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