arXiv subjects

Federico Bernini

Publications and source records attributed to Federico Bernini.

9 recordsLinked to original sources

arXiv2026

Qualitative properties of the fractional magnetic $p$-Laplacian and applications to critical quasilinear problems

We investigate the fractional magnetic $p$-Laplacian operator in the physical dimension case $N=3$, with $0<s<1<p$ and $sp<3$. Our goal is twofold. First, we define and study suitable functional settings for such operator proving significant properties. Then we get the existence of weak solutions for some quasilinear equations involving a weighted critical and subcritical power type nonlinearity. Our technique relies on variational methods and faces various difficulties: the complex quasilinear framework due to the presence of an external magnetic potential, the nonlocal setting, which entails appropriate tools, and the lack of compactness, which requires concentration compactness arguments. In this direction, we state a new concentration compactness principle in the quasilinear magnetic setting that seems to be missing in the literature.

math.AP↗

arXiv2026

Existence and regularity for an entire Grushin-Choquard equation

We consider the following Choquard equation $$ -\Delta_\gamma u + u = \left(d(z)^{-\mu} \ast |u|^p\right)|u|^{p-2}u, \text{ in } \mathbb{R}^N, $$ where $\Delta_\gamma$ is the Grushin operator. For a suitable range of the parameter $p$ we prove the existence of a mountain pass solution of the equation and we establish that the solution belongs to $L^q(\mathbb{R}^N)$ for all $q\in [2,\infty]$ and to $C^{0,\alpha}_{\textrm{loc}}(\mathbb{R}^N)$ for some $\alpha \in (0,1)$. Additionally, we provide a Poho\v zaev type identity, which allows us to derive a nonexistence result for smooth solutions to our equation.

math.AP↗

arXiv2024

On a fractional magnetic pseudorelativistic operator: properties and applications

We introduce a fractional magnetic pseudorelativistic operator for a general fractional order $s\in(0,1)$. First we define a suitable functional setting and we prove some fundamental properties. Then we show the behavior of the operator as $s \nearrow 1$ obtaining some results \`a la Bourgain-Brezis-Mironescu and removing the singularity from the integral definition. Finally we get existence of weak solutions for some semilinear equations involving a power type nonlinearity or a nonlocal (Choquard type) term.

math.AP↗

arXiv2024

Multiplicity of critical orbits to nonlinear, strongly indefinite functionals with sign-changing nonlinear part

We show an abstract critical point theorem about existence of infinitely many critical orbits to strongly indefinite functionals with sign-changing nonlinear part defined on a dislocation space with a discrete group action. We apply the abstract result to a Schr\"odinger equation $$ -\Delta u + V(x) u = f(u) - \lambda g(u) $$ with $0$ in the spectral gap of the Schr\"odinger operator $-\Delta + V(x)$, that appears in nonlinear optics, as well as to the equations with singular potentials arising from the study of cylindrically symmetric, electromagnetic waves to the system of Maxwell equations.

math.AP↗

arXiv2024

Note on homoclinic solutions to nonautonomous Hamiltonian systems with sign-changing nonlinear part

In the paper, we utilize the recent variational, abstract theorem to show the existence of homoclinic solutions to the Hamiltonian system $$ \dot{z} = J D_z H(z, t), \quad t \in \mathbb{R}, $$ where the Hamiltonian $H : \mathbb{R}^{2N} \times \mathbb{R} \rightarrow \mathbb{R}$ is of the form $$ H(z, t) = \frac12 Az \cdot z + \Gamma(t) \left( F(z) - \lambda G(z) \right) $$ for some symmetric matrix $A$.

math.CA↗

arXiv2024

Nonlinear Schr\"odinger-Poisson systems in dimension two: the zero mass case

We provide an existence result for a Schr\"odinger-Poisson system in gradient form, set in the whole plane, in the case of zero mass. Since the setting is limiting for the Sobolev embedding, we admit nonlinearities with subcritical or critical growth in the sense of Trudinger-Moser. In particular, the absence of the mass term requires a nonstandard functional framework, based on homogeneous Sobolev spaces. These features, combined with the logarithmic behaviour of the kernel of the Poisson equation, make the analysis delicate, since standard variational tools cannot be applied. The system is solved by considering the corresponding logarithmic Choquard equation. The existence of a mountain pass-type solution is established by means of a careful analysis of appropriate Cerami sequences, whose boundedness is ensured through a nonstandard variational method, suggested by the subtle nature of the functional geometry involved. As a key tool in our estimates, we also introduce a logarithmic weighted Trudinger-Moser inequality, along with a related Cao-type inequality, both of which hold in our functional setting and are, we believe, of independent interest.

math.AP↗

arXiv2023

Compact embeddings for weighted fractional Sobolev spaces and applications to Nonlinear Schrödinger Equations

The aim of this work is to prove a compact embedding for a weighted fractional Sobolev spaces. As an application, we use this embedding to prove, via variational methods, the existence of solutions for the following Schrödinger equation $$ (-Δ)^su + V(|x|)u = K(|x|)f(u), \quad \text{ in } \mathbb{R}^N, $$ where the two measurable functions $K > 0$ and $V \geq 0$ could vanish at infinity.

math.AP↗

arXiv2022

Generalized linking-type theorem with applications to strongly indefinite problems with sign-changing nonlinearities

We show the linking-type result which allows us to study strongly indefinite problems with sign-changing nonlinearities. We apply the abstract theory to the singular Schrödinger equation $$ -Δu + V(x)u + \frac{a}{r^2} u = f(u) - λg(u), \quad x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \ r = |y|, $$ where $$ 0 \not\in σ\left( -Δ+ \frac{a}{r^2} + V(x) \right). $$ As a consequence we obtain also the existence of solutions to the nonlinear curl-curl problem.

math.AP↗

arXiv2021

Semirelativistic Choquard equations with singular potentials and general nonlinearities arising from Hartree-Fock theory

We are interested in the general Choquard equation \begin{multline*} \sqrt{\strut -Δ+ m^2} \ u - mu + V(x)u - \fracμ{|x|} u = \left( \int_{\mathbb{R}^N} \frac{F(y,u(y))}{|x-y|^{N-α}} \, dy \right) f(x,u) - K (x) |u|^{q-2}u \end{multline*} under suitable assumptions on the bounded potential $V$ and on the nonlinearity $f$. Our analysis extends recent results by the second and third author on the problem with $μ= 0$ and pure-power nonlinearity $f(x,u)=|u|^{p-2}u$. We show that, under appropriate assumptions on the potential, whether the ground state does exist or not. Finally, we study the asymptotic behaviour of ground states as $μ\to 0^+$.

math.AP↗