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Bartosz Bieganowski

Publications and source records attributed to Bartosz Bieganowski.

At least 37 records · Page 2Linked to original sources
arXiv2025

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 $$ H(z, t) = \frac12 Az \cdot z + Γ(t) \left( F(z) - λG(z) \right) $$ for some symmetric matrix $A$.

math.CA↗
arXiv2025

Semiclassical states for the curl-curl problem

We show the existence of the so-called semiclassical states $\mathbf{U}:\mathbb{R}^3\to\mathbb{R}^3$ to the following curl-curl problem $$ \varepsilon^2\; \nabla \times (\nabla \times \mathbf{U}) + V(x) \mathbf{U} = g(\mathbf{U}), $$ for sufficiently small $\varepsilon > 0$. We study the asymptotic behaviour of solutions as $\varepsilon\to 0^+$ and we investigate also a related nonlinear Schrödinger equation involving a singular potential. The problem models large permeability nonlinear materials satisfying the system of Maxwell equations.

math.AP↗
arXiv2024

Supervised Autoencoders with Fractionally Differentiated Features and Triple Barrier Labelling Enhance Predictions on Noisy Data

This paper investigates the enhancement of financial time series forecasting with the use of neural networks through supervised autoencoders (SAE), to improve investment strategy performance. Using the Sharpe and Information Ratios, it specifically examines the impact of noise augmentation and triple barrier labeling on risk-adjusted returns. The study focuses on Bitcoin, Litecoin, and Ethereum as the traded assets from January 1, 2016, to April 30, 2022. Findings indicate that supervised autoencoders, with balanced noise augmentation and bottleneck size, significantly boost strategy effectiveness. However, excessive noise and large bottleneck sizes can impair performance.

q-fin.TR↗
arXiv2024

Normalized solutions to at least mass critical problems: singular polyharmonic equations and related curl-curl problems

We are interested in the existence of normalized solutions to the problem \begin{equation*} \begin{cases} (-Δ)^m u+\fracμ{|y|^{2m}}u + λu = g(u), \quad x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \\ \int_{\mathbb{R}^N} |u|^2 \, dx = ρ> 0, \end{cases} \end{equation*} in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the $L^2$-ball. Moreover, we find also a solution to the related curl-curl problem \begin{equation*} \begin{cases} \nabla\times\nabla\times\mathbf{U}+λ\mathbf{U}=f(\mathbf{U}), \quad x \in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|\mathbf{U}|^2\,dx=ρ, \end{cases} \end{equation*} which arises from the system of Maxwell equations and is of great importance in nonlinear optics.

math.AP↗
arXiv2024

Supervised Autoencoder MLP for Financial Time Series Forecasting

This paper investigates the enhancement of financial time series forecasting with the use of neural networks through supervised autoencoders, aiming to improve investment strategy performance. It specifically examines the impact of noise augmentation and triple barrier labeling on risk-adjusted returns, using the Sharpe and Information Ratios. The study focuses on the S&P 500 index, EUR/USD, and BTC/USD as the traded assets from January 1, 2010, to April 30, 2022. Findings indicate that supervised autoencoders, with balanced noise augmentation and bottleneck size, significantly boost strategy effectiveness. However, excessive noise and large bottleneck sizes can impair performance, highlighting the importance of precise parameter tuning. This paper also presents a derivation of a novel optimization metric that can be used with triple barrier labeling. The results of this study have substantial policy implications, suggesting that financial institutions and regulators could leverage techniques presented to enhance market stability and investor protection, while also encouraging more informed and strategic investment approaches in various financial sectors.

q-fin.TR↗
arXiv2023

Magnetostatic levitation and two related linear PDEs in unbounded domains

We consider a problem occurring in a magnetostatic levitation. The problem leads to a linear PDE in a strip. In engineering literature a particular solution is obtained. Such a solution enables one to compute lift and drag forces of the levitating object. It is in agreement with the experiment. We show that such a solution is unique in a class of bounded regular functions. Moreover, as a byproduct, we obtain nonstandard uniqueness results in two linear PDEs in unbounded domains. One of them is an Eigenvalue problem for the Laplacian in the strip in the nonstandard class of functions.

math.AP↗
arXiv2022

Elliptic problems with mixed nonlinearities and potentials singular at the origin and at the boundary of the domain

We are interested in the following Dirichlet problem $$ \left\{ \begin{array}{ll} -Δu + λu - μ\frac{u}{|x|^2} - ν\frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus Ω)^2} = f(x,u) & \quad \mbox{in } Ω\\ u = 0 & \quad \mbox{on } \partial Ω, \end{array} \right. $$ on a bounded domain $Ω\subset \mathbb{R}^N$ with $0 \in Ω$. We assume that the nonlinear part is superlinear on some closed subset $K \subset Ω$ and asymptotically linear on $Ω\setminus K$. We find a solution with the energy bounded by a certain min-max level, and infinitely many solutions provided that $f$ is odd in $u$. Moreover we study also the multiplicity of solutions to the associated normalized problem.

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

Bound states for the Schrödinger equation with mixed-type nonlinearites

We prove the existence results for the Schrödinger equation of the form $$ -Δu + V(x) u = g(x,u), \quad x \in \mathbb{R}^N, $$ where $g$ is superlinear and subcritical in some periodic set $K$ and linear in $\mathbb{R}^N \setminus K$ for sufficiently large $|u|$. The periodic potential $V$ is such that $0$ lies in a spectral gap of $-Δ+V$. We find a solution with the energy bounded by a certain min-max level, and infinitely many geometrically distinct solutions provided that $g$ is odd in $u$.

math.AP↗
arXiv2021

Solutions to a nonlinear Maxwell equation with two competing nonlinearities in $\mathbb{R}^3$

We are interested in the nonlinear, time-harmonic Maxwell equation $$ \nabla \times (\nabla \times \mathbf{E} ) + V(x) \mathbf{E} = h(x, \mathbf{E})\mbox{ in } \mathbb{R}^3 $$ with sign-changing nonlinear term $h$, i.e. we assume that $h$ is of the form $$ h(x, αw) = f(x, α) w - g(x, α) w $$ for $w \in \mathbb{R}^3$, $|w|=1$ and $α\in \mathbb{R}$. In particular, we can consider the nonlinearity consisting of two competing powers $h(x, \mathbf{E}) = |\mathbf{E}|^{p-2}\mathbf{E} - |\mathbf{E}|^{q-2}\mathbf{E}$ with $2 < q < p < 6$. Under appriopriate assumptions, we show that weak, cylindrically equivariant solutions of the special form are in one-to-one correspondence with weak solutions to a Schrödinger equation with a singular potential. Using this equivalence result we show the existence of the least energy solution among cylindrically equivariant solutions of the particular form to the Maxwell equation, as well as to the Schrödinger equation.

math.AP↗
arXiv2021

Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth

We propose a simple minimization method to show the existence of least energy solutions to the normalized problem \begin{cases} -Δu + λu = g(u) \quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, \\ u \in H^1(\mathbb{R}^N), \\ \int_{\mathbb{R}^N} |u|^2 \, dx = ρ> 0, \end{cases} where $ρ$ is prescribed and $(λ, u) \in \mathbb{R} \times H^1 (\mathbb{R}^N)$ is to be determined. The new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints is demonstrated, which allows to provide general growth assumptions imposed on $g$. We cover the most known physical examples and nonlinearities with growth considered in the literature so far as well as we admit the mass critical growth at $0$.

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↗
arXiv2019

Non-local to local transition for ground states of fractional Schrödinger equations on bounded domains

We show that ground state solutions to the nonlinear, fractional problem \begin{align*} \left\{ \begin{array}{ll} (-Δ)^{s} u + V(x) u = f(x,u) &\quad \mathrm{in} \ Ω, \newline u = 0 &\quad \mathrm{in} \ \mathbb{R}^N \setminus Ω, \end{array} \right. \end{align*} on a bounded domain $Ω\subset \mathbb{R}^N$, converge (along a subsequence) in $L^2 (Ω)$, under suitable conditions on $f$ and $V$, to a solution of the local problem as $s \to 1^-$.

math.AP↗
arXiv2018

Schrödinger-type equations with sign-changing nonlinearities: a survey

We are looking for solutions to nonlinear Schrödinger-type equations of the form $$ (-Δ)^{α/ 2} u (x) + V(x) u(x) = h (x,u(x)), \quad x \in \mathbb{R}^N, $$ where $V : \mathbb{R}^N \rightarrow \mathbb{R}$ is an external potential (bounded, unbounded or singular) and the nonlinearity $h : \mathbb{R}^N \times \mathbb{R} \rightarrow \mathbb{R}$ may be sign-changing. The aim of this paper is to give an introduction to the problem and the variational approach based on the Nehari manifold technique, and to survey recent results concerning the existence of ground and bound states.

math.AP↗
arXiv2018

Note on semiclassical states for the Schrödinger equation with nonautonomous nonlinearities

We consider the following Schrödinger equation $$ - \hslash ^2 Δu + V(x)u = Γ(x) f(u) \quad \mathrm{in} \ \mathbb{R}^N, $$ where $u \in H^1 (\mathbb{R}^N)$, $u > 0$, $\hslash > 0$ and $f$ is superlinear and subcritical nonlinear term. We show that if $V$ attains local minimum and $Γ$ attains global maximum at the same point or $V$ attains global minimum and $Γ$ attains local maximum at the same point, then there exists a positive solution for sufficiently small $\hslash>0$.

math.AP↗
arXiv2018

Systems of coupled Schrödinger equations with sign-changing nonlinearities via classical Nehari manifold approach

We propose existence and multiplicity results for the system of Schrödinger equations with sign-changing nonlinearities in bounded domains or in the whole space $\mathbb{R}^N$. In the bounded domain we utilize the classical approach via the Nehari manifold, which is (under our assumptions) a differentiable manifold of class $\mathcal{C}^1$ and the Fountain theorem by Bartsch. In the space $\mathbb{R}^N$ we additionally need to assume the $\mathbb{Z}^N$-periodicity of potentials and our proofs are based on the concentration-compactness lemma by Lions and the Lusternik-Schnirelmann values.

math.AP↗
arXiv2018

The fractional Schrödinger equation with Hardy-type potentials and sign-changing nonlinearities

We look for solutions to a fractional Schrödinger equation of the following form $$ (-Δ)^{α/ 2} u + \left( V(x) - \fracμ{|x|^α} \right) u = f(x,u)-K(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N \setminus \{0\}, $$ where $V$ is bounded and close-to-periodic potential and $- \fracμ{|x|^α}$ is a Hardy-type potential. We assume that $V$ is positive and $f$ has the subcritical growth but not higher than $|u|^{q-2}u$. If $μ$ is positive and small enough we find a ground state solution, i.e. a critical point of the energy being minimizer on the Nehari manifold. If $μ$ is negative we show that there is no ground state solutions. We are also interested in an asymptotic behaviour of solutions as $μ\to 0^+$ and $K \to 0$.

math.AP↗