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

We discuss some recent results dealing with the existence of bound states of the nonlinear Schrodinger-Poisson system

 $$\left\{ \begin{gathered} - \Delta u + V(x)u + \lambda K(x)\phi (x)u = |u|^{{p - 1}} u, \hfill \\ - \Delta \phi = K(x)u^{2}, \hfill \\ \end{gathered} \right.$$
 as well as of the corresponding semiclassical limits. The proofs are based upon Critical Point theory and Perturbation Methods.

On Schrödinger-Poisson Systems

In this paper, we consider the following Schrodinger–Poisson system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u=f(u)&{}\quad \text{ in }\ \mathbb {R}^3,\\ -\Delta \phi =u^2&{}\quad \text{ in }\ \mathbb {R}^3. \end{array} \right. \end{aligned}$$
 We investigate the existence of multiple bound state solutions, in particular sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions. In particular, the nonlinear term includes the power-type nonlinearity $$f(u)=|u|^{p-2}u$$
 for the well-studied case $$p\in (4,6)$$
 , and the less studied case $$p\in (3,4)$$
 , and for the latter case, few existence results are available in the literature.

/pdf/infinitely-many-sign-changing-solutions-for-the-nonlinear-ic90kkjlw5.pdf

Infinitely many sign-changing solutions for the nonlinear Schrödinger–Poisson system

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

Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential

Let $(M,g)$ be a non-locally conformally flat compact Riemannian manifold with dimension $N\ge7.$ We are interested in finding positive solutions to the linear perturbation of the Yamabe problem $$-\mathcal L_g u+\epsilon u=u^{N+2\over N-2}\ \hbox{in}\ (M,g) $$ where the first eigenvalue of the conformal laplacian $-\mathcal L_g $ is positive and $\epsilon$ is a small positive parameter. We prove that for any point $\xi_0\in M$ which is non-degenerate and non-vanishing minimum point of the Weyl's tensor and for any integer $k$ there exists a family of solutions developing $k$ peaks collapsing at $\xi_0$ as $\epsilon$ goes to zero. In particular, $\xi_0$ is a non-isolated blow-up point.

Clustering phenomena for linear perturbation of the Yamabe equation

Let (M, g) be a compact smooth connected Riemannian manifold (without boundary) of dimension N ≥ 7. Assume M is symmetric with respect to a point ξ0 with non-vanishing Weyl’s tensor. We consider the linear perturbation of the Yamabe problem 

$$ (P_{\epsilon })\qquad -\mathcal {L}_{g} u+\epsilon u=u^{\frac {N+2}{N-2}}\ \text { in }\ (M,g) . $$

We prove that for any k ∈ ℕ, there exists ek > 0 such that for all e ∈ (0, ek) the problem (P𝜖) has a symmetric solution ue, which looks like the superposition of k positive bubbles centered at the point ξ0 as e → 0. In particular, ξ0 is a towering blow-up point.

Towering Phenomena for the Yamabe Equation on Symmetric Manifolds

In this paper we study a system which is equivalent to a nonlocal version of the well known Brezis Nirenberg problem. The difficulties related with the lack of compactness are here emphasized by the nonlocal nature of the critical nonlinear term. We prove existence and nonexistence results of positive solutions when $N=3$ and existence of solutions in both the resonance and the nonresonance case for higher dimensions.

Generalized Schr\"odinger-Newton system in dimension $N\ge 3$: critical case

/pdf/bubbling-solutions-for-supercritical-problems-on-manifolds-4ai6rqpgfp.pdf

Giusi Vaira

Papers

Clustering phenomena for linear perturbation of the Yamabe equation

Towering Phenomena for the Yamabe Equation on Symmetric Manifolds

Generalized Schr\"odinger-Newton system in dimension $N\ge 3$: critical case

Bubbling solutions for supercritical problems on manifolds

On the Stability for Paneitz-Type Equations