We construct solutions to a class of Schrodinger equations involving the fractional Laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.

Ground state solutions for nonlinear fractional Schrödinger equations in RN

Ground state solutions for the nonlinear Schrödinger–Maxwell equations

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 prove the existence of a nontrivial solution 𝑢 ∈ H¹ (ℝ^N) to the nonlinear Choquard equation -Δ 𝑢 + 𝑢 = (I_α * 𝐹 (𝑢)) 𝐹' (𝑢) in ℝ^N, where I_α is a Riesz potential, under almost necessary conditions on the nonlinearity 𝐹 in the spirit of Berestycki and Lions. This solution is a groundstate; if moreover 𝐹 is even and monotone on (0, ∞), then 𝑢 is of constant sign and radially symmetric.

/pdf/existence-of-groundstates-for-a-class-of-nonlinear-choquard-3v56oiaoi2.pdf

Existence of groundstates for a class of nonlinear Choquard equations

Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3

Minimum action solutions for a quasilinear equation

In this paper we prove the existence of a positive solution to the equation $-\Delta u + V(x)u=g(u)$ in $R^N,$ assuming the general hypotheses on the nonlinearity introduced by Berestycki & Lions. Moreover we show that a minimizing problem, related to the existence of a ground state, has no solution.

On the Schrodinger equation in $R^N$ under the effect of a general nonlinear term

In this paper we study a system which we propose as a model to describe the interaction between matter and electromagnetic field from a dualistic point of view. This system arises from a suitable coupling of the Schrodinger and the Born–Infeld agrangians, this latter replacing the role that, classically, is played by the Maxwell Lagrangian. We use a variational approach to find an electrostatic radial ground state solution by means of suitable estimates on the functional of the action.

On the Schrödinger-Born-Infeld system

Generalized Schrödinger–Newton system in dimension N ⩾ 3: Critical case

In this paper we prove the existence of a positive energy static solution for the Chern–Simons–Schrodinger system under a large-distance fall-off requirement on the gauge potentials. We are also interested in existence of ground state solutions.

/pdf/positive-energy-static-solutions-for-the-chern-simons-56lod14lpd.pdf

Antonio Azzollini

Papers

Minimum action solutions for a quasilinear equation

On the Schrodinger equation in $R^N$ under the effect of a general nonlinear term

On the Schrödinger-Born-Infeld system

Generalized Schrödinger–Newton system in dimension N ⩾ 3: Critical case

Positive energy static solutions for the Chern–Simons–Schrödinger system under a large-distance fall-off requirement on the gauge potentials