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Bartosz Bieganowski

Publications and source records attributed to Bartosz Bieganowski.

At least 19 recordsLinked to original sources
arXiv2026

Existence of solutions for a large defocusing parameter in a strongly indefinite Schrödinger equation

Let $N\geq 3$, $2 0$ such that, for every $λ\geqλ_\infty$, the competing-power equation $$ -Δu+V(x)u=|u|^{p-2}u-λ|u|^{q-2}u \quad\text{in }\mathbb{R}^N $$ has a nontrivial solution $u_λ\in H^1(\mathbb{R}^N)$. Moreover, $$ \|u_λ\|_{H^1(\mathbb{R}^N)}+\|u_λ\|_{L^\infty(\mathbb{R}^N)} \leq Cλ^{-1/(q-2)}. $$ We also show that, after the natural rescaling and lattice translations, a sequence of such solutions converges weakly to a nontrivial solution of the pure defocusing equation $Lv=-|v|^{q-2}v$.

math.AP↗
arXiv2026

A generalisation of the Gagliardo--Nirenberg Inequality with applications to mass-critical and mass-subcritical elliptic equations

Via a new inequality à la Gagliardo--Nirenberg, we prove the existence and nonexistence of solutions to \begin{equation*} \begin{cases} (-Δ)^s u + \fracμ{|y|^{2s}} u + λu = f(u), \quad \mathbb{R}^N \ni x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \\ \int_{\mathbb{R}^N} u^2 \, \mathrm{d}x = ρ\end{cases} \end{equation*} in the mass-critical and mass-subcritical regimes, where $s>0$, $N \ge K \ge 2$, $μ\in \mathbb{R}$ belongs to a specific range, $ρ>0$ is given a priori, and $λ\in \mathbb{R}$ is unknown. Additionally, we obtain similar results for the problem above with $μ=0$ and $N \ge 1$ as well as a related curl-curl equation. Finally, we provide a thorough insight into the threshold for $ρ$ that divides the scenarios of negative and zero least energy.

math.AP↗
arXiv2026

Ground states for strongly indefinite Schrödinger equations with competing nonlinearities

We survey recent variational methods for strongly indefinite Schrödinger equations with sign-changing nonlinearities. The main object is an energy functional of the form \[ J(u)=\frac12\|u^+\|^2-\frac12\|u^-\|^2 -\int_{\mathbb{R}^N}F(u)\,dx+λ\int_{\mathbb{R}^N}G(u)\,dx, \] where the splitting $X=X^+\oplus X^-$ is induced by a spectral gap of the linear Schrödinger operator, and where the nonlinear part \[ I(u)=\int_{\mathbb{R}^N}F(u)\,dx-λ\int_{\mathbb{R}^N}G(u)\,dx \] is allowed to change sign. We discuss the generalized linking theorem developed for such functionals, and the abstract multiplicity theory for critical orbits in dislocation spaces. In the final part, we prove a new ground state result for the competing pure-power case \[ f(u)=|u|^{p-2}u,\qquad g(u)=|u|^{q-2}u,\qquad 2 0$ sufficiently small, the corresponding strongly indefinite functional possesses a nontrivial critical point of least energy among all nontrivial critical points.

math.AP↗
arXiv2026

Nonlinear Schrödinger equations with a critical, inverse-square potential

We study the existence of solutions of the following nonlinear Schrödinger equation $$ -Δu+V(x)u-\frac{(N-2)^2}{4|x|^2}u=f(x,u) $$ where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ are periodic with respect to $x\in\mathbb{R}^N.$ We assume that $V$ has positive essential infimum, $f$ satisfies weak growth conditions and $N\geq 3$. The approach to the problem uses variational methods with nonstandard functional setting. We obtain the existence of the ground state solution using the new profile decomposition.

math.AP↗
arXiv2026

Finite-energy solutions to Einstein-scalar field Lichnerowicz equations on complete Riemannian manifolds

We consider the singular elliptic problem of the form \[ -Δu + V(x)u = \mathcal{B}(x)|u|^{2^*-2}u + \frac{\mathcal{A}(x)}{|u|^{2^*}u}, \qquad u\in H^1(M), \] where the coefficients are allowed to have low regularity. Under natural spectral assumptions on $-Δ+V$, geometric assumptions on the manifold $M$ ensuring the Sobolev embedding $H^1(M)\hookrightarrow L^{2^*}(M)$, and a suitable global integrability/smallness condition involving $\mathcal{A}$, $\mathcal{B}$, and a function $ψ\in H^1(M)$, we prove the existence of a nonnegative finite-energy supersolution. If, in addition, the Ricci curvature is nonnegative and $\mathcal{B}\ge 0$, we obtain a positive finite-energy solution. The proof relies on a family of $\varepsilon$-regularized problems, mountain pass arguments, and a limiting procedure in which Harnack's inequality plays a crucial role in handling the singular term on noncompact manifolds. We also prove a nonexistence result showing that the global integrability condition on $\mathcal{A}$ is, in a precise sense, necessary for the existence of nonnegative supersolutions.

math.AP↗
arXiv2026

Putnam-like dataset summary: LLMs as mathematical competition contestants

In this paper we summarize the results of the Putnam-like benchmark published by Google DeepMind. This dataset consists of 96 original problems in the spirit of the Putnam Competition and 576 solutions generated by LLMs. We analyze the performance of models on this set of problems to verify their ability to solve problems from mathematical contests. We find that top models, particularly Gemini 2.5 Pro, achieve high scores, demonstrating strong mathematical reasoning capabilities, although their performance was lower on problems from the 2024 Putnam competition. The analysis highlights distinct behavioral patterns among models, including bimodal scoring distributions and challenges in providing fully rigorous justifications.

cs.LG↗
arXiv2026

Explainable Patterns in Cryptocurrency Microstructure

We document stable cross-asset patterns in cryptocurrency limit-order-book microstructure: the same engineered order book and trade features exhibit remarkably similar predictive importance and SHAP dependence shapes across assets spanning an order of magnitude in market capitalization (BTC, LTC, ETC, ENJ, ROSE). The data covers Binance Futures perpetual contract order books and trades on 1-second frequency starting from January 1st, 2022 up to October 12th, 2025. Using a unified CatBoost modeling pipeline with a direction-aware GMADL objective and time-series cross validation, we show that feature rankings and partial effects are stable across assets despite heterogeneous liquidity and volatility. We connect these SHAP structures to microstructure theory (order flow imbalance, spread, and adverse selection) and validate tradability via a conservative top-of-book taker backtest as well as fixed depth maker backtest. Our primary novelty is a robustness analysis of a major flash crash, where the divergent performance of our taker and maker strategies empirically validates classic microstructure theories of adverse selection and highlights the systemic risks of algorithmic trading. Our results suggest a portable microstructure representation of short-horizon returns and motivate universal feature libraries for crypto markets.

q-fin.TR↗
arXiv2026

Nonlinear scalar field equations with a critical Hardy potential

We study the existence of solutions for the nonlinear scalar field equation $$-Δu - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\},$$ where the potential $-\frac{(N-2)^2}{4|x|^2}$ is the critical Hardy potential and $N \geq 3$. The nonlinearity $g$ is continuous and satisfies general subcritical growth assumptions of the Berestycki-Lions type. The problem is approached using variational methods within a non-standard functional setting. The natural energy functional associated with the equation is defined on the space $X^1(\mathbb{R}^N)$, which is the completion of $H^1(\mathbb{R}^N)$ with respect to the norm induced by the quadratic part of the functional. We establish the existence of a nontrivial solution $u_0 \in X^1(\mathbb{R}^N)$ that satisfies the Pohožaev constraint $\mathcal{M}$ and minimizes the energy functional on $\mathcal{M}$. Furthermore, assuming $g$ is odd, we prove the existence of at least one non-radial solution.

math.AP↗
arXiv2025

Ground state solutions to the nonlinear Born-Infeld problem

In the paper we show the existence of ground state solutions to the nonlinear Born-Infeld problem \[ \mathrm{div}\, \left( \frac{\nabla u}{\sqrt{1-|\nabla u|^2}} \right) + f(u) = 0, \quad x \in \mathbb{R}^N \] in the zero and positive mass cases. Moreover, we find a new proof of the Sobolev-type inequality \[ \int_{\mathbb{R}^N} \left(1 - \sqrt{1-|\nabla u|^2}\right) \, dx \geq C_{N,p} \left( \int_{\mathbb{R}^N} |u|^p \, dx \right)^{\frac{N}{N+p}}, \] for $p > 2^*$ as well as the characterization of the optimal constant $C_{N,p}$ in terms of the ground state energy level. Previous approaches relied on approximation schemes and/or symmetry assumptions, which typically yield to compact embeddings and may lead to solutions that are not at the ground state energy level. In contrast, neither approximation arguments nor symmetry assumptions are employed in the paper to obtain a ground state solution. Instead, we develop a new direct variational approach based on minimization over a Pohožaev manifold combined with profile decomposition techniques. Finally, we show that nonradial solutions exist whenever $N \geq 4$; in particular, this settles a previously open problem in the case $N=5$.

math.AP↗
arXiv2025

A Lichnerowicz equation in the Einstein-scalar field theory on non-CMC closed manifolds

In the paper, we prove the existence of a positive and essentially bounded solution to a Lichnerowicz equation in the Einstein-scalar field theory on a closed manifold with non-constant mean curvature. In particular, the non-constant mean curvature gives rise to supercritical terms in the equation, on top of singular ones. We employ a recent fixed-point argument, which involves sub- and supersolutions. Additionally, we provide several conditions on the coefficients in the equation that prevent the existence of positive classical solutions.

math.AP↗
arXiv2025

Normalized solutions to polyharmonic equations with Hardy-type potentials and exponential critical nonlinearities

Via a constrained minimization, we find a solution $(λ,u)$ to the problem \begin{equation*} \begin{cases} (-Δ)^m u+\fracμ{|x|^{2m}}u + λu = ηu^3 + g(u)\\ \int_{\mathbb{R}^{2m}} u^2 \, dx = ρ\end{cases} \end{equation*} with $1 \le m \in \mathbb{N}$, $μ,η\ge 0$, $ρ> 0$, and $g$ having exponential critical growth at infinity and mass supercritical growth at zero.

math.AP↗
arXiv2025

Traveling waves for nonlinear Schrödinger equations

We look for traveling wave solutions to the nonlinear Schrödinger equation with a subsonic speed, covering several physical models with Sobolev subcritical nonlinear effects. Our approach is based on a variant of Sobolev-type inequality involving the momentum and we show the existence of its minimizers solving the nonlinear Schrödinger equation.

math.AP↗
arXiv2025

Existence and dynamics of normalized solutions to Schrödinger equations with generic double-behaviour nonlinearities

We study the existence of solutions $(\underline u,λ_{\underline u})\in H^1(\mathbb{R}^N; \mathbb{R}) \times \mathbb{R}$ to \[ -Δu + λu = f(u) \quad \text{in } \mathbb{R}^N \] with $N \ge 3$ and prescribed $L^2$ norm, and the dynamics of the solutions to \[ \begin{cases} \mathrm{i} \partial_t Ψ+ ΔΨ= f(Ψ)\\ Ψ(\cdot,0) = ψ_0 \in H^1(\mathbb{R}^N; \mathbb{C}) \end{cases} \] with $ψ_0$ close to $\underline u$. Here, the nonlinear term $f$ has mass-subcritical growth at the origin, mass-supercritical growth at infinity, and is more general than the sum of two powers. Under different assumptions, we prove the existence of a locally least-energy solution, the orbital stability of all such solutions, the existence of a second solution with higher energy, and the strong instability of such a solution.

math.AP↗
arXiv2025

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 Schrödinger equation $$ -Δu + V(x) u = f(u) - λg(u) $$ with $0$ in the spectral gap of the Schrödinger operator $-Δ+ V(x)$, 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↗
arXiv2025

Time-zero limits of Kaden's spirals and 2D Euler

The present note is devoted to the studies of the relation of the time-zero limits of Kaden's spirals and the 2D Euler equation. It is shown that the time-zero limits of Kaden's spirals satisfy inhomogeneous 2D Euler in a weak sense. As a corollary, the necessity of both, the decay of spherical averages around the origin of the spiral as well as the velocity matching condition, for the 2D Euler equation to hold in a weak sense, is shown. Finally, some preliminary results concerning the Kaden spirals are obtained.

math.AP↗
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↗