We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations 

$$\begin{aligned} -\Delta u + V(x)u = \left( |x|^{-(N-\alpha )} *|u |^p\right) |u |^{p - 2} u \quad \text {in} \ \mathbb {R}^N, \end{aligned}$$

and some of its variants and extensions.

/pdf/a-guide-to-the-choquard-equation-4ufiqlw479.pdf

A guide to the Choquard equation

Nodal solutions for the Choquard equation

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$ 
-\Delta u + V(x)u = \bigl(|x|^{-(N-\alpha)} * |u|^p\bigr)|u|^{p - 2} u \qquad \text{in $\mathbb{R}^N$}, $$ and some of its variants and extensions.

/pdf/a-guide-to-the-choquard-equation-5eemtjk1tb.pdf

We consider nonlinear Choquard equation where N ≥ 3, V ∈ L∞(ℝN) is an external potential and Iα(x) is the Riesz potential of order α ∈ (0, N). The power in the nonlocal part of the equation is critical with respect to the Hardy–Littlewood–Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if then the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis–Lieb lemma.

Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent

We study the nonlocal equation −e²Δue + Vue = e−α (Iα∗∣ue∣p)∣ue∣p−2ue in ℝN, where N≥1, α∈(0,N), Iα(x)=Aα/∣x∣N−α is the Riesz potential and e>0 is a small parameter. We show that if the external potential V∈C(ℝN;[0,∞)) has a local minimum and p∈[2,(N+α)/(N−2)+) then for all small e>0 the problem has a family of solutions concentrating to the local minimum of V provided that: either p>1+max(α,α+22)/(N−2)+, or p>2 and lim inf ∣x∣→∞ V(x)∣x∣2>0, or p=2 and inf x∈ℝN V(x)(1+∣x∣N−α)>0. Our assumptions on the decay of V and admissible range of p≥2 are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work.

Semi-classical states for the Choquard equation

We develop a nonlinear quantum theory of Newtonian gravity consistent with an objective interpretation of the wavefunction. Inspired by the ideas of Schrodinger, and Bell, we seek a dimensional reduction procedure to map complex wavefunctions in configuration space onto a family of observable fields in space-time. Consideration of quasi-classical conservation laws selects the reduced one-body quantities as the basis for an explicit quasi-classical coarse-graining. These we interpret as describing the objective reality of the laboratory. Thereafter, we examine what may stand in the role of the usual Copenhagen observer to localise this quantity against macroscopic dispersion. Only a tiny change is needed, via a generically attractive self-potential. A nonlinear treatment of gravitational self-energy is thus advanced. This term sets a scale for all wavepackets. The Newtonian cosmology is thus closed, without need of an external observer. Finally, the concept of quantisation is re-interpreted as a nonlinear eigenvalue problem. To illustrate, we exhibit an elementary family of gravitationally self-bound solitary waves. Contrasting this theory with its canonically quantised analogue, we find that the given interpretation is empirically distinguishable, in principle. This result encourages deeper study of nonlinear field theories as a testable alternative to canonically quantised gravity.

/pdf/newtonian-quantum-gravity-pa1908dk7d.pdf

Newtonian quantum gravity

Puts forward a complete scenario for interpreting nonlinear field theories highlighting the role played by gravitational self--energy in enabling a consistent revival of the Schroedinger approach to unifying micro and macro physics.

/pdf/newtonian-quantum-gravity-3kq7jmcrcz.pdf

K. R. W. Jones

Papers

Newtonian quantum gravity

Newtonian Quantum Gravity