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

Positive solutions for some non-autonomous Schrödinger–Poisson systems

Abstract We study the existence of semiclassical states for a nonlinear Schrödinger-Poisson system that concentrate near critical points of the external potential and of the density charge function. We use a perturbation scheme in a variational setting, extending the results in [1]. We also discuss necessary conditions for concentration.

On Concentration of Positive Bound States for the Schrödinger-Poisson Problem with Potentials

In this paper we consider the following elliptic system in $${\mathbb{R}^3}$$
 $$\qquad\left\{\begin{array}{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\end{array}\right.$$
 where λ is a real parameter, $${p\in (1, 5)}$$
 if λ < 0 while $${p\in (3, 5)}$$
 if λ > 0 and K(x), a(x) are non-negative real functions defined on $${\mathbb{R}^3}$$
 . Assuming that $${\lim_{|x|\rightarrow+\infty}K(x)=K_{\infty} >0 }$$
 and $${\lim_{|x|\rightarrow+\infty}a(x)=a_{\infty} >0 }$$
 and satisfying suitable assumptions, but not requiring any symmetry property on them, we prove the existence of positive ground states, namely the existence of positive solutions with minimal energy.

Ground states for Schrödinger–Poisson type systems

In this paper we consider the system in $\R^3$ \label{problemadipartenza0} -\e^2\Delta u+V(x)u+\phi(x)u=u^{p},

Cluster solutions for the Schrodinger-Poisson-Slater problem around a local minimum of the potential

In this paper we consider the system in $\mathbb{R}^3$ \begin{equation} \left\{ \begin{array}{l} -\varepsilon^2 \Delta u + V(x)u + \phi(x)u = u^p, \\ -\Delta \phi = u^2, \end{array} \right. \end{equation} for $p\in (1,5)$. We prove the existence of multi-bump solutions whose bumps concentrate around a local minimum of the potential $V(x)$. We point out that such solutions do not exist in the framework of the usual Nonlinear Schrodinger Equation.

https://projecteuclid.org/download/pdf_1/euclid.rmi/1296828834

Giusi Vaira

Papers

Positive solutions for some non-autonomous Schrödinger–Poisson systems

On Concentration of Positive Bound States for the Schrödinger-Poisson Problem with Potentials

Ground states for Schrödinger–Poisson type systems

Cluster solutions for the Schrodinger-Poisson-Slater problem around a local minimum of the potential

Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential