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 a multiplicity result concerning the critical points of a class of functionals involving local and nonlocal nonlinearities. We apply our result to the nonlinear Schrodinger-Maxwell system and to the nonlinear elliptic Kirchhoff equation assuming on the local nonlinearity the general hypotheses introduced by Berestycki and Lions.

Multiple critical points for a class of nonlinear functionals

In this paper we study a model which describes the relation of the matter and the electromagnetic field from a unitarian standpoint in the spirit of the ideas of Born and Infeld. This model, introduced in [1], is based on a semilinear perturbation of the Maxwell equation (SME). The particles are described by the finite energy solitary waves of SME whose existence is due to the presence of the nonlinearity. In the magnetostatic case (i.e. when the electric field ${\bf E}=0$
 and the magnetic field ${\bf H}$
 does not depend on time) the semilinear Maxwell equations reduce to semilinear equation 
where “
 $\nabla\times $
 ” is the curl operator, f′ is the gradient of a smooth function $f:{\mathbb{R}}^3\to{\mathbb{R}}$
 and ${\bf A}:{\mathbb{R}}^3\to{\mathbb{R}}^3$
 is the gauge potential related to the magnetic field ${\bf H}$
 (
 ${\bf H}=\nabla\times {\bf A}$
 ). The presence of the curl operator causes (1) to be a strongly degenerate elliptic equation. The existence of a nontrivial finite energy solution of (1) having a kind of cylindrical symmetry is proved. The proof is carried out by using a variational approach based on two main ingredients: the Principle of symmetric criticality of Palais, which allows to avoid the difficulties due to the curl operator, and the concentration-compactness argument combined with a suitable minimization argument. Keywords: Maxwell equations, Natural constraint, Minimizing sequence Mathematics Subject Classification (2000): 35B40, 35B45, 92C15

/pdf/existence-of-static-solutions-of-the-semilinear-maxwell-14mw74g8db.pdf

Existence of static solutions of the semilinear Maxwell equations

Ground state solution for a problem with mean curvature operator in Minkowski space

In this paper we prove the existence of a nontrivial non-negative radial solution for the quasilinear elliptic problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\nabla \cdot \left[\phi ^{\prime }(|\nabla u|^2)\nabla u \right] +|u|^{\alpha -2}u =|u|^{s-2} u,&x\in \mathbb{R }^{N},\\ u(x) \rightarrow 0, \quad \text{ as} |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$
 where $$N\ge 2, \phi (t)$$
 behaves like $$t^{q/2}$$
 for small $$t$$
 and $$t^{p/2}$$
 for large $$t, 1< p<q<N, 1<\alpha \le p^* q^{\prime }/p^{\prime }$$
 and $$\max \{q,\alpha \}< s<p^*,$$
 being $$p^*=\frac{pN}{N-p}$$
 and $$p^{\prime }$$
 and $$q^{\prime }$$
 the conjugate exponents, respectively, of $$p$$
 and $$q$$
 . Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.

Quasilinear elliptic equations in \mathbb{R }^{N} via variational methods and Orlicz-Sobolev embeddings

We study the class of nonlinear Klein-Gordon-Maxwell systems describing a standing wave (charged matter field) in equilibrium with a purely electrostatic field. We improve some previous existence results in the case of an homogeneous nonlinearity. Moreover, we deal with a limit case, namely when the frequency of the standing wave is equal to the mass of the charged field; this case shows analogous features of the well known "zero mass case" for scalar field equations.

Antonio Azzollini

Papers

Multiple critical points for a class of nonlinear functionals

Existence of static solutions of the semilinear Maxwell equations

Ground state solution for a problem with mean curvature operator in Minkowski space

Quasilinear elliptic equations in \mathbb{R }^{N} via variational methods and Orlicz-Sobolev embeddings

Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system