arXiv subjects

Jaeyoung Byeon

Publications and source records attributed to Jaeyoung Byeon.

5 recordsLinked to original sources

arXiv2026

Nonlinear Schr\"odinger systems with all attractive forces

In this paper we investigate in a systematic way the solution structure of nonnegative solutions for coupled nonlinear Schr\"odinger systems in the all attractive regime. For all frequencies equal case we obtain results on Morse index of synchronized positive vector solutions and nonnegative semi-vector solutions as well as their kernel of linearized systems at these solutions. We establish a variational characterization of synchronized positive vector solutions and as applications we give a new existence result about positive vector solutions for the general systems with arbitrary frequencies. We also examine the synchronization phenomenon of positive vector solutions, and prove that if the interaction matrix has exactly one positive eigenvalue, any positive vector solution is synchronized. Finally we provide examples of domains for which synchronization of positive solutions fails to hold if the interaction matrix has at least two positive eigenvalues. Our results reveal the effect of the spectral information of the interaction matrix of the couplings and geometry of a domain on the solution structure.

math.AP↗

arXiv2024

Compactness via monotonicity in nonsmooth critical point theory, with application to Born-Infeld type equations

In this paper, we prove new existence and multiplicity results for critical points of lower semicontinuous functionals in Banach spaces, complementing the nonsmooth critical point theory set forth by Szulkin and avoiding the need of the Palais-Smale condition. We apply our abstract results to get entire solutions with finite energy to Born-Infeld type autonomous equations. More precisely, under almost optimal conditions on the nonlinearity, we construct a positive solution and infinitely many solutions both in the classes of radially symmetric functions and nonradiallly symmetric ones.

math.AP↗

arXiv2023

Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born-Infeld model

Given a measure $ρ$ on a domain $Ω\subset \mathbb{R}^m$, we study spacelike graphs over $Ω$ in Minkowski space with Lorentzian mean curvature $ρ$ and Dirichlet boundary condition on $\partial Ω$. The graph function $u_ρ: Ω\rightarrow \mathbb{R}$ also represents the electric potential generated by a charge $ρ$ in electrostatic Born-Infeld theory. While $u_ρ$ minimizes the action $$ I_ρ(ψ) = \int_Ω \Big( 1 - \sqrt{1-|Dψ|^2} \Big) \mathrm{d} x - \langle ρ, ψ\rangle $$ among competitors with $|Dψ| \le 1$, because of a lack of smoothness of the Lagrangian density when $|Dψ| = 1$ a direct approach via minimization may not produce a solution to the Euler-Lagrange equation (BI). In this paper, we study existence and regularity of $u_ρ$ for general $ρ$, in a bounded domain and in the entire $\mathbb{R}^m$. In particular, we find sufficient conditions to guarantee that $u_ρ$ solves (BI) and enjoys log-improved $W^{2,2}_{\mathrm{loc}}$ estimates, and we construct examples helping to identify sharp thresholds for the regularity of $ρ$ to ensure the validity of (BI). One of the main difficulties is the possible presence of light segments in the graph of $u_ρ$, which will be discussed in detail.

math.AP↗

arXiv2018

Hardy's inequality in a limiting case on general bounded domains

In this paper, we study Hardy's inequality in a limiting case: $$ \int_Ω |\nabla u |^N dx \ge C_N(Ω) \int_Ω \frac{|u(x)|^N}{|x|^N \left(\log \frac{R}{|x|} \right)^N} dx $$ for functions $u \in W^{1,N}_0(Ω)$, where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $R = \sup_{x \in Ω} |x|$. We study the (non-)attainability of the best constant $C_N(Ω)$ in several cases. We provide sufficient conditions that assure $C_N(Ω) > C_N(B_R)$ and $C_N(Ω)$ is attained, here $B_R$ is the $N$-dimensional ball with center the origin and radius $R$. Also we provide an example of $Ω\subset \mathbb{R}^2$ such that $C_2(Ω) > C_2(B_R) = 1/4$ and $C_2(Ω)$ is not attained.

math.AP↗

arXiv2008

Symmetry and monotonicity of least energy solutions

We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable.

math.AP↗