The Choquard equation in \(\mathbb{R}^N\) writes as \[ -\Delta u + V u = (I_\alpha \ast u^p)u^{p - 1}, \] where \(I_\alpha : \mathbb{R}^n \to \mathbb{R}\) is a Riesz potential defined for \(x \in \mathbb{R}^n \setminus \{0\}\) by \[ I_\alpha (x) = \frac{A_\alpha}{\vert x \vert^{N - \alpha}}. \] This equation was introduced in 1976 by P. Choquard in order to describe an electron trapped in its own hole. It also appears in the theory of the polaron at rest and in models of interaction between non relativistic quantum mechanics and gravitation.
For this equation,
- with V. Moroz, we have given nonexistence results for supersolutions in outer domains,
- with V. Moroz, in the autonomous case \(V = 1\), we have studied properties of solutions and existence for general nonlinearities, the corresponding problem in the plane was considered in collaboration with L. Battaglia,
- with V. Moroz, we have studied associated nonlocal Hardy inequalities,
- with V. Moroz, we have constructed solutions in the semi-classical limit,
- with Xia Jiankang, we have studied the case of confining potentials and the semi-classicali limit in the critical frequency case,
- with V. Moroz, we have studied the existence for the
lower critical exponent \(p = 1 + \frac{\alpha}{N}\), - with V. Moroz and M. Ghimenti, we have studied the existence of minimal action nodal solutions,
- with D. Ruiz, we have studied the odd symmetry of these minimal action nodal solutions,
- with D. Bonheure and S. Cingolani, we have proved the nondegeneracy of the solution to the two-dimensional logarithmic Choquard equation.
