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On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents. (English) Zbl 1357.35106

Summary: We consider the following nonlinear Choquard equation with Dirichlet boundary condition \[ -\Delta u = \left( \int_{\Omega} \frac{| u |^{2_\mu^\ast}}{| x - y |^\mu} dy \right) | u |^{2_\mu^\ast - 2} u + \lambda f(u) \text{ in } \Omega, \] where \(\Omega\) is a smooth bounded domain of \(\mathbb{R}^N\), \(\lambda > 0\), \(N \geq 3\), \(0 < \mu < N\) and \(2_\mu^\ast\) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on different types of nonlinearities \(f(u)\), we are able to prove some existence and multiplicity results for the equation by variational methods.

MSC:

35J20 Variational methods for second-order elliptic equations
35B33 Critical exponents in context of PDEs
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