A guide to 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.Comment: 39 page

Cytotoxic T lymphocytes directed against a tumor-specific mutated antigen display similar HLA tetramer binding but distinct functional avidity and tissue distribution

We have previously identified an antigen (Ag) recognized on a human large cell carcinoma of the lung by a tumor-specific cytotoxic T lymphocyte clone derived from autologous tumor infiltrating lymphocytes (TILs). The antigenic peptide is presented by HLA-A2 molecules and is encoded by a mutated α-actinin-4 (ACTN4) gene. In the present report, we have isolated two anti-α-actinin-4 T cell clones from the same patient TIL and from his peripheral blood lymphocytes (PBLs) by using tetramers of soluble HLA-A2 molecules loaded with the mutated peptide. Although all of the clones displayed similar tetramer labeling, those isolated from PBL showed lower avidity of Ag recognition and killed the specific target much less efficiently, indicating that tetramer staining does not correlate with clone avidity/tumor reactivity. T cell receptor (TCR) analysis revealed that α-actinin-4-reactive clones used distinct α and β chain rearrangements, demonstrating TCR repertoire diversity. Interestingly, TCRβ chain gene usage indicated that only Ag-specific clones with high functional avidity were expanded at the tumor site, whereas a low-avidity clone was exclusively amplified in patient peripheral blood. Our results point to the existence of distinct but overlapping antitumor TCR repertoires in TIL and PBL and suggest a selective in situ expansion of tumor-specific cytotoxic T lymphocyte with high avidity/tumor reactivity

Semi-classical states for the Choquard equation

We study the nonlocal equation $$-\varepsilon^2 \Delta u_\varepsilon + V u_\varepsilon = \varepsilon^{-\alpha} \bigl(I_\alpha \ast \lvert u_\varepsilon\rvert^p\bigr) \lvert u_\varepsilon \rvert^{p - 2} u_\varepsilon\quad\text{in \(\mathbf{R}^N\)},$$ where $N \ge 1$, $\alpha \in (0, N)$, $I_\alpha (x) = A_\alpha/\lvert x \rvert^{N - \alpha}$ is the Riesz potential and $\varepsilon > 0$ is a small parameter. We show that if the external potential $V \in C (\mathbb{R}^N; [0, \infty))$ has a local minimum and $p \in [2, (N + \alpha)/(N - 2)_+)$ then for all small $\varepsilon > 0$ the problem has a family of solutions concentrating to the local minimum of $V$ provided that: either $p > 1 + \max (\alpha, \frac{\alpha + 2}{2})/(N - 2)_+$, or $p > 2$ and $\liminf_{\lvert x\rvert \to \infty} V (x) \lvert x \rvert^2 > 0$, or $p = 2$ and $\inf_{x \in \mathbb{R}^N} V (x) (1 + \lvert x \rvert^{N - \alpha}) > 0$. Our assumptions on the decay of $V$ and admissible range of $p\ge 2$ are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work.Comment: 28 pages, updated bibliograph