OPUSCULA MATHEMATICA

Permanent URI for this communityhttps://repo.agh.edu.pl/handle/AGH/102755

Adres wydawniczy: Kraków : Wydawnictwa AGH, 2005-
O czasopiśmie: http://www.opuscula.agh.edu.pl/
ISSN: 1232-9274 e-ISSN: 2300-6919
DOI: http://dx.doi.org/10.7494/OpMath

The journal Opuscula Mathematica publishes original research articles that are of significant importance in all areas of Discrete Mathematics, Functional Analysis, Differential Equations, Mathematical Physics, Nonlinear Analysis, Numerical Analysis, Probability Theory and Statistics, Theory of Optimal Control and Optimization, Financial Mathematics and Mathematical Economic Theory, Operations Research, and other areas of Applied Mathematics.

New! Aktualny numer: 2026 - Vol. 46 - No. 3

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Opuscula Mathematica
2026 - Vol. 46 - No. 3
Access status: Open Access ,
Existence of solutions for a doubly critical Schrödinger-Poisson system on the first Heisenberg group
(Wydawnictwa AGH, 2026) Ma, Xueyan; Shi, Shaoyun; Song, Yueqiang
This work is devoted to the study of a class of Schrödinger-Poisson system with doubly critical growth on the first Heisenberg group. Utilizing the concentration-compactness principle associated with classical Sobolev space on the Heisenberg group and mountain pass theorem, we prove that the system admits multiple nontrivial solutions.
Access status: Open Access ,
On a relation between growth estimates and Harnack inequalities for quasilinear elliptic equations with nonlinear lower order terms
(Wydawnictwa AGH, 2026) Hirata, Kentaro
We investigate a relation between the Harnack inequalities and the (a priori) growth estimates for positive solutions of quasilinear elliptic equations with nonlinear terms involving the solution and its gradient in an arbitrary domain in $\mathbb{R}^N$.
Access status: Open Access ,
A priori estimates and existence of positive solutions for elliptic problems under integral Neumann boundary conditions
(Wydawnictwa AGH, 2026) Corrêa, Francisco J.S.A.; Figueiredo, Giovany M.; Morbach, Joelma
In this paper, we establish a priori estimates and existence of positive solutions for elliptic problems under integral Neumann boundary conditions.
Access status: Open Access ,
Comparison theorems for property (B) of the third-order differential equations with deviating arguments
(Wydawnictwa AGH, 2026) Džurina, Jozef; Baculíková, Blanka
The aim of this paper is to introduce a new comparison theorem (in both delayed and advanced cases) that allows us to investigate the properties of third-order differential equations with quasi-derivatives $(r_{1}(t)(r_{2}(t)y'(t))')'-p(t)y(\tau(t))=0$ using the following simpler differential equations $(r(t)(r(t)z'(t))')'-p(t)z(\tau(t))=0$ and $y'''(t)-q(t)y(\sigma(t))=0.$ The obtained comparison principles allow for the immediate transcription of the oscillatory results known for the simpler equations into studied equation with quasi-derivatives. The progress achieved will be illustrated through several examples.
Access status: Open Access ,
Normalized ground states for a $p$-Laplacian system in the mass super-critical case
(Wydawnictwa AGH, 2026) Tao, Yuhang; Zhang, Jianjun
In this paper, we study the existence of positive normalized solutions to the following $p$-Laplacian system: $\begin{cases} -\Delta_p u+\lambda_1u^{p-1}=\mu_1u^{m_1-1}+\beta r_1u^{r_1-1}v^{r_2}&\text{in }\mathbb{R}^N,\\ -\Delta_p v+\lambda_2v^{p-1}=\mu_2v^{m_2-1}+\beta r_2u^{r_1}v^{r_2-1}&\text{in }\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^p=a, \quad \int_{\mathbb{R}^N}|v|^p=b,\end{cases}$ where $1\lt p\lt N$, $\mu_1,\mu_2,\beta,a,b\gt 0$ are prescribed, $\lambda_1,\lambda_2 \in \mathbb{R}$ are known as the Lagrange multiplier, $\Delta_p u= \mathrm{div} (|\nabla u|^{p-2} \nabla u)$ denotes the $p$-Laplacian operator. We prove the existence of positive solutions for the coupled purely mass super-critical case (i.e., $\frac{p^2}{N}+p\lt m_1,m_2,r_1 + r_2\lt p^*$) by a minimization argument based on a closed ball and the Pohozaev constraint.
Access status: Open Access ,
On mixed local-nonlocal Sobolev-type inequalities and their connection with singular equations in the Heisenberg group
(Wydawnictwa AGH, 2026) Garain, Prashanta
In this work, we establish a mixed local-nonlocal Sobolev-type inequality in the Heisenberg group and demonstrate that its extremals coincide with solutions to the corresponding mixed local-nonlocal singular $p$-Laplace equations. We further show that these inequalities serve as a necessary and sufficient condition for the existence of weak solutions to the associated singular problems. Notably, the same characterization remains valid in both the purely local and purely nonlocal settings. Our results thus provide a unified framework linking the existence theory for singular equations across local, nonlocal, and mixed regimes.
Access status: Open Access ,
Normalized solutions for planar Schrödinger-Poisson system with critical exponential growth and nonlocal interaction
(Wydawnictwa AGH, 2026) Wei, Chenlu; Chen, Sitong; Shu, Muhua
This paper focuses on the following planar Schrödinger-Poisson system with critical exponential growth and nonlocal interaction $\begin{cases}-\Delta u+\lambda u+\mu(\log|\cdot|*u^2)u = \gamma \left( I_\alpha * |u|^q \right) |u|^{q-2} u+\left(e^{u^2}-1-u^2\right)u, & x\in \mathbb{R}^2, \\ \displaystyle \int_{\mathbb{R}^2}u^2\mathrm{d}x=c,\end{cases}$ where $c\gt 0$, $\mu,\gamma\gt 0$, $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier, $\alpha \in (0,2)$, $1+\frac{\alpha}{2} \leq q \lt +\infty$, $I_\alpha:\mathbb{R}^2\to\mathbb{R}$ denotes the Riesz potential and $1+\frac{\alpha}{2}$ is the lower critical exponent with respect to the Hardy-Littlewood-Sobolev inequality. Through delicate energy estimates, under explicit conditions on $c$, we prove the existence of two normalized solutions: one is a local minimizer and the other is of mountain-pass type. The presence of the logarithmic kernel and the competition between the two nonlocal terms necessitates the development of new tools to address the loss of compactness caused by the critical exponential growth, for which the variational techniques developed for the local problem are no longer applicable. Our work not only generalizes the special case $\gamma=0$, but also provides an analytical approach that is applicable to more $L^2$-constrained problems with competing nonlocal terms modelling long-range attraction in particle physics.
Access status: Open Access ,
On the existence of independent (1,k) -dominating sets for k∈{1,2} in two products of graphs
(Wydawnictwa AGH, 2026) Bednarz, Paweł; Michalski, Adrian; Paja, Natalia
A subset $J$ of vertices is said to be a $(1,k)$-dominating set if every vertex $v$ not belonging to the set $J$ has a neighbour in $J$ and there exists also another vertex in $J$ within the distance at most $k$ from $v$. In this paper, we study the problem of the existence of independent $(1,k)$-dominating sets for $k\in\{1,2\}$ in the tensor product and in the strong product of two graphs. We give complete characterisations of these graph products, which have independent $(1,1)$-dominating sets or independent $(1,2)$-dominating sets, with respect to the properties of their factors.
Access status: Open Access ,
Solutions with prescribed mass for a critical Choquard equation driven by a local-nonlocal operator
(Wydawnictwa AGH, 2026) Nidhi, Nidhi; Sreenadh, Konijeti
In this paper, we study the normalized solutions of the following critical growth Choquard equation with mixed local and nonlocal operators: \[\begin{split}-\Delta u +(-\Delta)^s u &= \lambda u +\mu |u|^{p-2}u +(I_{\alpha}*|u|^{2^*_{\alpha}})|u|^{2^*_{\alpha}-2}u \quad\text{in}\quad \mathbb{R}^N,\\ \| u\|_2 &= \tau,\end{split}\] where $N\geq 3$, $\tau\gt 0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (0,N)$, $2^*_{\alpha}=\frac{N+\alpha}{N-2}$ is the critical exponent corresponding to the Hardy-Littlewood-Sobolev inequality, $(-\Delta)^s$ is the nonlocal fractional Laplacian operator with $s\in (0,1)$, $\mu\gt 0$ is a parameter and $\lambda$ appears as Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of the mass-subcritical perturbation $\mu |u|^{p-2}u$ with $2\gt p\gt 2+\frac{4s}{N}$ under some assumptions on $\tau$.

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