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

In this paper we prove the existence of a nontrivial solution to the nonlinear Schrodinger-Maxwell equations in $\R^3,$ assuming on the nonlinearity the general hypotheses introduced by Berestycki & Lions.

On the Schrodinger-Maxwell equations under the effect of a general nonlinear term

In this paper we prove the existence of a nontrivial solution to the nonlinear Schrodinger–Maxwell equations in R 3 , assuming on the nonlinearity the general hypotheses introduced by Berestycki and Lions.

/pdf/on-the-schrodinger-maxwell-equations-under-the-effect-of-a-3udheyc83r.pdf

On the Schrödinger-Maxwell equations under the effect of a general nonlinear term

In this paper we prove the existence of a ground state solution for the nonlinear Klein–Gordon–Maxwell equations in the electrostatic case.

/pdf/ground-state-solutions-for-the-nonlinear-klein-gordon-4yazdb99mq.pdf

Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations

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

Antonio Azzollini

Papers

Ground state solutions for the nonlinear Schrödinger–Maxwell equations

On the Schrodinger-Maxwell equations under the effect of a general nonlinear term

On the Schrödinger-Maxwell equations under the effect of a general nonlinear term

Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations

On the Schrödinger Equation in RN under the effect of a general nonlinear term