Nonlinear Schr\"odinger equation in the Bopp-Podolsky electrodynamics: solutions in the electrostatic case

We study the following nonlinear Schr\"odinger-Bopp-Podolsky system $$\begin{cases} -\Delta u + \omega u + q^{2}\phi u = |u|^{p-2}u -\Delta \phi + a^2 \Delta^2 \phi = 4\pi u^2 \end{cases} \hbox{ in }\mathbb{R}^3$$ with $a,\omega>0$. We prove existence and nonexistence results depending on the parameters $q,p$. Moreover we also show that, in the radial case, the solutions we find tend to solutions of the classical Schr\"odinger-Poisson system as $a\to0$.Comment: 30 pages, the nonexistence result has been improve

Nonlinear fractional magnetic Schr\"odinger equation: existence and multiplicity

In this paper we focus our attention on the following nonlinear fractional Schr\"odinger equation with magnetic field \begin{equation*} \varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N\geq 3$, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $A:\mathbb{R}^{N}\rightarrow \mathbb{R}^N$ are continuous potentials and $f:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a subcritical nonlinearity. By applying variational methods and Ljusternick-Schnirelmann theory, we prove existence and multiplicity of solutions for $\varepsilon$ small.Comment: 23 page

Soliton dynamics for the Schrodinger-Newton system

We investigate the soliton dynamics for the Schrodinger-Newton system by proving a suitable modulational stability estimates in the spirit of those obtained by Weinstein for local equations.Comment: 10 page

Ground states for fractional magnetic operators

We study a class of minimization problems for a nonlocal operator involving an external magnetic potential. The notions are physically justified and consistent with the case of absence of magnetic fields. Existence of solutions is obtained via concentration compactness.Comment: 22 pages, minor corrections and typos fixe

Quasilinear elliptic equations in \RN via variational methods and Orlicz-Sobolev embeddings

In this paper we prove the existence of a nontrivial non-negative radial solution for a quasilinear elliptic problem. 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.Comment: 18 pages, 1 figur

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

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.Comment: 18 pages, typos fixed, some minor revision