A list of articles of the latest volume. 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. https://www.opuscula.agh.edu.pl AGH University of Science and Technology Press en Opuscula Mathematica 1232-9274 2300-6919 https://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg https://www.opuscula.agh.edu.pl 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. https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4619.pdf Normalized ground states for a p-Laplacian system in the mass super-critical case Yuhang Tao Jianjun Zhang \(p\)-Laplacian system, positive normalized solution, coupled purely mass super-critical case doi:10.7494/OpMath.202603311 Opuscula Math. 46, no. 3 (2026), 405-434, https://doi.org/10.7494/OpMath.202603311 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202603311 https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4619.pdf 46 3 405 434
Title: Normalized ground states for a p-Laplacian system in the mass super-critical case.

Author(s): Yuhang Tao, Jianjun Zhang.

Abstract: 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.
Keywords: \(p\)-Laplacian system, positive normalized solution, coupled purely mass super-critical case.
Mathematics Subject Classification: 35J47, 35J62.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 3 (2026), 405-434, https://doi.org/10.7494/OpMath.202603311.

 

]]> 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. https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4618.pdf Normalized solutions for planar Schrödinger-Poisson system with critical exponential growth and nonlocal interaction Chenlu Wei Sitong Chen Muhua Shu normalized solution, logarithmic convolution potential, nonlocal interaction, critical exponential growth, Trudinger-Moser inequality doi:10.7494/OpMath.202605041 Opuscula Math. 46, no. 3 (2026), 367-404, https://doi.org/10.7494/OpMath.202605041 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202605041 https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4618.pdf 46 3 367 404
Title: Normalized solutions for planar Schrödinger-Poisson system with critical exponential growth and nonlocal interaction.

Author(s): Chenlu Wei, Sitong Chen, Muhua Shu.

Abstract: 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.
Keywords: normalized solution, logarithmic convolution potential, nonlocal interaction, critical exponential growth, Trudinger-Moser inequality.
Mathematics Subject Classification: 35J20, 35J62, 35Q55.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 3 (2026), 367-404, https://doi.org/10.7494/OpMath.202605041.

 

]]> 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. https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4617.pdf Existence of solutions for a doubly critical Schrödinger-Poisson system on the first Heisenberg group Xueyan Ma Shaoyun Shi Yueqiang Song Heisenberg group, Schrödinger-Poisson system, concentration-compactness principle, mountain pass theorem, nontrivial solutions doi:10.7494/OpMath.202605211 Opuscula Math. 46, no. 3 (2026), 347-366, https://doi.org/10.7494/OpMath.202605211 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202605211 https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4617.pdf 46 3 347 366
Title: Existence of solutions for a doubly critical Schrödinger-Poisson system on the first Heisenberg group.

Author(s): Xueyan Ma, Shaoyun Shi, Yueqiang Song.

Abstract: 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.
Keywords: Heisenberg group, Schrödinger-Poisson system, concentration-compactness principle, mountain pass theorem, nontrivial solutions.
Mathematics Subject Classification: 35A15, 35B33, 47G20.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 3 (2026), 347-366, https://doi.org/10.7494/OpMath.202605211.

 

]]> 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 \(\mathbb{R}^N\). https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4616.pdf On a relation between growth estimates and Harnack inequalities for quasilinear elliptic equations with nonlinear lower order terms Kentaro Hirata growth estimate, Harnack inequality, quasilinear elliptic equation doi:10.7494/OpMath.202603272 Opuscula Math. 46, no. 3 (2026), 323-345, https://doi.org/10.7494/OpMath.202603272 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202603272 https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4616.pdf 46 3 323 345
Title: On a relation between growth estimates and Harnack inequalities for quasilinear elliptic equations with nonlinear lower order terms.

Author(s): Kentaro Hirata.

Abstract: 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 \(\mathbb{R}^N\).
Keywords: growth estimate, Harnack inequality, quasilinear elliptic equation.
Mathematics Subject Classification: 35J92, 35B09, 35B45.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 3 (2026), 323-345, https://doi.org/10.7494/OpMath.202603272.

 

]]> 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. https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4615.pdf On mixed local-nonlocal Sobolev-type inequalities and their connection with singular equations in the Heisenberg group Prashanta Garain Sobolev type inequality, extremal, mixed local-nonlocal singular problem, Heisenberg group doi:10.7494/OpMath.202603271 Opuscula Math. 46, no. 3 (2026), 305-321, https://doi.org/10.7494/OpMath.202603271 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202603271 https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4615.pdf 46 3 305 321
Title: On mixed local-nonlocal Sobolev-type inequalities and their connection with singular equations in the Heisenberg group.

Author(s): Prashanta Garain.

Abstract: 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.
Keywords: Sobolev type inequality, extremal, mixed local-nonlocal singular problem, Heisenberg group.
Mathematics Subject Classification: 35A23, 35H20, 35J92, 35R11, 35J75.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 3 (2026), 305-321, https://doi.org/10.7494/OpMath.202603271.

 

]]> 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. https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4614.pdf Comparison theorems for property (B) of the third-order differential equations with deviating arguments Jozef Džurina Blanka Baculíková canonical equation, comparison theorem, property (B) doi:10.7494/OpMath.202605181 Opuscula Math. 46, no. 3 (2026), 291-304, https://doi.org/10.7494/OpMath.202605181 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202605181 https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4614.pdf 46 3 291 304
Title: Comparison theorems for property (B) of the third-order differential equations with deviating arguments.

Author(s): Jozef Džurina, Blanka Baculíková.

Abstract: 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.
Keywords: canonical equation, comparison theorem, property (B).
Mathematics Subject Classification: 34K11, 34C10.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 3 (2026), 291-304, https://doi.org/10.7494/OpMath.202605181.

 

]]> In this paper, we establish a priori estimates and existence of positive solutions for elliptic problems under integral Neumann boundary conditions. https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4613.pdf A priori estimates and existence of positive solutions for elliptic problems under integral Neumann boundary conditions Francisco J.S.A. Corrêa Giovany M. Figueiredo Joelma Morbach a priori estimates, positive solution, integral Neumann boundary condition doi:10.7494/OpMath.202605201 Opuscula Math. 46, no. 3 (2026), 273-290, https://doi.org/10.7494/OpMath.202605201 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202605201 https://www.opuscula.agh.edu.pl/vol46/3/art/opuscula_math_4613.pdf 46 3 273 290
Title: A priori estimates and existence of positive solutions for elliptic problems under integral Neumann boundary conditions.

Author(s): Francisco J.S.A. Corrêa, Giovany M. Figueiredo, Joelma Morbach.

Abstract: In this paper, we establish a priori estimates and existence of positive solutions for elliptic problems under integral Neumann boundary conditions.
Keywords: a priori estimates, positive solution, integral Neumann boundary condition.
Mathematics Subject Classification: 35A15, 45P05.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 3 (2026), 273-290, https://doi.org/10.7494/OpMath.202605201.

 

]]> 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 \(\tau\). https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4612.pdf Solutions with prescribed mass for a critical Choquard equation driven by a local-nonlocal operator Nidhi Nidhi Konijeti Sreenadh normalized solution, Choquard equation, critical exponent, mixed local and nonlocal operator, \(L^2\)-subcritical perturbation, nonlinear Schrödinger equation driven by local-nonlocal operator doi:10.7494/OpMath.202603181 Opuscula Math. 46, no. 2 (2026), 235-266, https://doi.org/10.7494/OpMath.202603181 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202603181 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4612.pdf 46 2 235 266
Title: Solutions with prescribed mass for a critical Choquard equation driven by a local-nonlocal operator.

Author(s): Nidhi Nidhi, Konijeti Sreenadh.

Abstract: 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 \(\tau\).
Keywords: normalized solution, Choquard equation, critical exponent, mixed local and nonlocal operator, \(L^2\)-subcritical perturbation, nonlinear Schrödinger equation driven by local-nonlocal operator.
Mathematics Subject Classification: 35Q55, 35M10, 35J61, 35A01.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 235-266, https://doi.org/10.7494/OpMath.202603181.

 

]]> We provide an alternative approach, based on the Leray-Schauder fixed point index in cones, to a fixed point theorem for operator systems due to Precup. Our focus is on the case of operators whose components are entirely of compressive type. The abstract technique is applied to a system of second-order differential equations providing a coexistence positive solution by means of an eigenvalue type criterion. https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4611.pdf On a fixed point theorem for operator systems and eigenvalue criteria for existence of positive solutions Laura M. Fernández-Pardo Jorge Rodríguez-López coexistence fixed point, fixed point index, positive solution, nonlinear systems doi:10.7494/OpMath.202602271 Opuscula Math. 46, no. 2 (2026), 219-234, https://doi.org/10.7494/OpMath.202602271 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202602271 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4611.pdf 46 2 219 234
Title: On a fixed point theorem for operator systems and eigenvalue criteria for existence of positive solutions.

Author(s): Laura M. Fernández-Pardo, Jorge Rodríguez-López.

Abstract: We provide an alternative approach, based on the Leray-Schauder fixed point index in cones, to a fixed point theorem for operator systems due to Precup. Our focus is on the case of operators whose components are entirely of compressive type. The abstract technique is applied to a system of second-order differential equations providing a coexistence positive solution by means of an eigenvalue type criterion.
Keywords: coexistence fixed point, fixed point index, positive solution, nonlinear systems.
Mathematics Subject Classification: 47H10, 47H11, 34B18, 34B16, 34C25.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 219-234, https://doi.org/10.7494/OpMath.202602271.

 

]]> We study the problem of finding a minimum \(k\)-critical-bipartite graph of order \((n,m)\): a bipartite graph \(G=(U,V;E)\), with \(|U|=n\), \(|V|=m\), and \(n\gt m\gt 1\), which is \(k\)-critical-bipartite, and the tuple \((|E|, \Delta_U, \Delta_V)\), where \(\Delta_U\) and \(\Delta_V\) denote the maximum degree in \(U\) and \(V\), respectively, is lexicographically minimum over all such graphs. \(G\) is \(k\)-critical-bipartite if deleting any set of at most \(k=n-m\) vertices from \(U\) yields \(G'\) that has a complete matching, i.e., a matching of size \(m\). Cichacz and Suchan solved the problem for biregular bipartite graphs. Here, we extend their results to bipartite graphs that are not biregular. We prove tight lower bounds on the connectivity of \(k\)-critical-bipartite graphs, and we show that \(k\)-critical-bipartite graphs are expander graphs. https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4610.pdf Minimum k-critical-bipartite graphs: the irregular case Sylwia Cichacz Agnieszka Görlich Karol Suchan fault-tolerance, interconnection network, bipartite graph, complete matching, algorithm, \(k\)-critical-bipartite graph doi:10.7494/OpMath.202602181 Opuscula Math. 46, no. 2 (2026), 201-218, https://doi.org/10.7494/OpMath.202602181 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202602181 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4610.pdf 46 2 201 218
Title: Minimum k-critical-bipartite graphs: the irregular case.

Author(s): Sylwia Cichacz, Agnieszka Görlich, Karol Suchan.

Abstract: We study the problem of finding a minimum \(k\)-critical-bipartite graph of order \((n,m)\): a bipartite graph \(G=(U,V;E)\), with \(|U|=n\), \(|V|=m\), and \(n\gt m\gt 1\), which is \(k\)-critical-bipartite, and the tuple \((|E|, \Delta_U, \Delta_V)\), where \(\Delta_U\) and \(\Delta_V\) denote the maximum degree in \(U\) and \(V\), respectively, is lexicographically minimum over all such graphs. \(G\) is \(k\)-critical-bipartite if deleting any set of at most \(k=n-m\) vertices from \(U\) yields \(G'\) that has a complete matching, i.e., a matching of size \(m\). Cichacz and Suchan solved the problem for biregular bipartite graphs. Here, we extend their results to bipartite graphs that are not biregular. We prove tight lower bounds on the connectivity of \(k\)-critical-bipartite graphs, and we show that \(k\)-critical-bipartite graphs are expander graphs.
Keywords: fault-tolerance, interconnection network, bipartite graph, complete matching, algorithm, \(k\)-critical-bipartite graph.
Mathematics Subject Classification: 05C35, 05C70, 05C85, 68M10, 68M15.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 201-218, https://doi.org/10.7494/OpMath.202602181.

 

]]> We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schrödinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. As a continuation of our previous study on the existence and uniqueness of solutions, we prove here some strong stabilization properties. In particular, we show the finite time extinction of solutions induced by the nonlinear saturation mechanism, which, sometimes, can be understood as a bang-bang control. The analysis relies on refined energy methods. Our results provide a rigorous justification of nonlinear dissipation as an effective stabilization mechanism for this class of complex equations where the maximum principle fails. https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4609.pdf Damped nonlinear Ginzburg-Landau equation with saturation. Part II. Strong Stabilization Pascal Bégout Jesús Ildefonso Díaz damped Ginzburg-Landau equation, saturated nonlinearity, finite time extinction doi:10.7494/OpMath.202603112 Opuscula Math. 46, no. 2 (2026), 185-199, https://doi.org/10.7494/OpMath.202603112 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202603112 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4609.pdf 46 2 185 199
Title: Damped nonlinear Ginzburg-Landau equation with saturation. Part II. Strong Stabilization.

Author(s): Pascal Bégout, Jesús Ildefonso Díaz.

Abstract: We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schrödinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. As a continuation of our previous study on the existence and uniqueness of solutions, we prove here some strong stabilization properties. In particular, we show the finite time extinction of solutions induced by the nonlinear saturation mechanism, which, sometimes, can be understood as a bang-bang control. The analysis relies on refined energy methods. Our results provide a rigorous justification of nonlinear dissipation as an effective stabilization mechanism for this class of complex equations where the maximum principle fails.
Keywords: damped Ginzburg-Landau equation, saturated nonlinearity, finite time extinction.
Mathematics Subject Classification: 35Q56, 35B40, 93D40.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 185-199, https://doi.org/10.7494/OpMath.202603112.

 

]]> We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schrödinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. Under suitable structural conditions on the complex coefficients, we establish the existence and uniqueness of global solutions. The analysis relies on the delicate proofs that the maximal monotone operator theory can be adapted to this framework, even for unbounded domains. https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4608.pdf Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains Pascal Bégout Jesús Ildefonso Díaz damped Ginzburg-Landau equation, saturated nonlinearity, finite time extinction, maximal monotone operators, existence and regularity of weak solutions doi:10.7494/OpMath.202603111 Opuscula Math. 46, no. 2 (2026), 153-183, https://doi.org/10.7494/OpMath.202603111 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202603111 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4608.pdf 46 2 153 183
Title: Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains.

Author(s): Pascal Bégout, Jesús Ildefonso Díaz.

Abstract: We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schrödinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. Under suitable structural conditions on the complex coefficients, we establish the existence and uniqueness of global solutions. The analysis relies on the delicate proofs that the maximal monotone operator theory can be adapted to this framework, even for unbounded domains.
Keywords: damped Ginzburg-Landau equation, saturated nonlinearity, finite time extinction, maximal monotone operators, existence and regularity of weak solutions.
Mathematics Subject Classification: 35Q56, 35A01, 35A02, 35D30, 35D35.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 153-183, https://doi.org/10.7494/OpMath.202603111.

 

]]> 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. https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4607.pdf On the existence of independent (1,k)-dominating sets for k\in\{1,2\} in two products of graphs Paweł Bednarz Adrian Michalski Natalia Paja dominating set, independent set, multiple domination, secondary domination, tensor product, strong product doi:10.7494/OpMath.202601201 Opuscula Math. 46, no. 2 (2026), 139-152, https://doi.org/10.7494/OpMath.202601201 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202601201 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4607.pdf 46 2 139 152
Title: On the existence of independent (1,k)-dominating sets for k\in\{1,2\} in two products of graphs.

Author(s): Paweł Bednarz, Adrian Michalski, Natalia Paja.

Abstract: 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.
Keywords: dominating set, independent set, multiple domination, secondary domination, tensor product, strong product.
Mathematics Subject Classification: 05C69, 05C76.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 139-152, https://doi.org/10.7494/OpMath.202601201.

 

]]> In this note, we prove that every graph can be edge-labelled with red labels \(1,2\) and blue labels \(1,2\) so that vertices with any sum of incident red labels induce a \(1\)-degenerate graph, while vertices with any sum of incident blue labels induce a \(0\)-degenerate graph. This result stands as a closer step towards the so-called Standard \((2,2)\)-Conjecture (stating that \(0\)-degeneracy can be achieved in both colours), and provides some insight on the surrounding field, which covers the 1-2-3 Conjecture, the 1-2 Conjecture, and other close problems. Along the way, we also describe how many related problems are interconnected, and raise new problems and questions for further work on the topic. https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4606.pdf Some remarks and results on the Standard (2,2)-Conjecture Olivier Baudon Julien Bensmail Lyn Vayssieres 1-2-3 Conjecture, 1-2 Conjecture, proper labelling, labelling doi:10.7494/OpMath.202602101 Opuscula Math. 46, no. 2 (2026), 127-137, https://doi.org/10.7494/OpMath.202602101 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202602101 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4606.pdf 46 2 127 137
Title: Some remarks and results on the Standard (2,2)-Conjecture.

Author(s): Olivier Baudon, Julien Bensmail, Lyn Vayssieres.

Abstract: In this note, we prove that every graph can be edge-labelled with red labels \(1,2\) and blue labels \(1,2\) so that vertices with any sum of incident red labels induce a \(1\)-degenerate graph, while vertices with any sum of incident blue labels induce a \(0\)-degenerate graph. This result stands as a closer step towards the so-called Standard \((2,2)\)-Conjecture (stating that \(0\)-degeneracy can be achieved in both colours), and provides some insight on the surrounding field, which covers the 1-2-3 Conjecture, the 1-2 Conjecture, and other close problems. Along the way, we also describe how many related problems are interconnected, and raise new problems and questions for further work on the topic.
Keywords: 1-2-3 Conjecture, 1-2 Conjecture, proper labelling, labelling.
Mathematics Subject Classification: 05C78, 05C15, 68R10.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 127-137, https://doi.org/10.7494/OpMath.202602101.

 

]]> This study focuses on a sequence of approximate minimizers for the functional \[J(u)=\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{p_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{p_{i}(x)}dx-\mu\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{q_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{q_{i}(x)}dx-\int\limits_{\Omega} F(u(x))dx,\] where \(\Omega\subset\mathbb{R}^N\) (\(N\geq 3\)) is a bounded domain, and \(p_i,q_i\in C(\overline{\Omega})\) with \(1\lt p_i,q_i\lt +\infty\) for all \(i \in \{1,\ldots,N\}\). We establish the convergence result to the infimum of \(J(u)\) when \(F:\mathbb{R}\to\mathbb{R}\) is a locally Lipschitz function of controlled growth, following the Galerkin method. As an application, we establish the existence of solutions to a class of Dirichlet inclusions associated to the functional. https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4605.pdf Galerkin-type minimizers to a competing problem for (\vec{p},\vec{q})-Laplacian with variable exponents Zhenfeng Zhang Mina Ghasemi Calogero Vetro anisotropic Sobolev space, Clarke generalized gradient, Dirichlet problem, Galerkin basis, \((\vec{p},\vec{q})\)-Laplacian with variable exponents doi:10.7494/OpMath.202511231 Opuscula Math. 46, no. 1 (2026), 101-119, https://doi.org/10.7494/OpMath.202511231 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202511231 https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4605.pdf 46 1 101 119
Title: Galerkin-type minimizers to a competing problem for (\vec{p},\vec{q})-Laplacian with variable exponents.

Author(s): Zhenfeng Zhang, Mina Ghasemi, Calogero Vetro.

Abstract: This study focuses on a sequence of approximate minimizers for the functional \[J(u)=\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{p_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{p_{i}(x)}dx-\mu\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{q_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{q_{i}(x)}dx-\int\limits_{\Omega} F(u(x))dx,\] where \(\Omega\subset\mathbb{R}^N\) (\(N\geq 3\)) is a bounded domain, and \(p_i,q_i\in C(\overline{\Omega})\) with \(1\lt p_i,q_i\lt +\infty\) for all \(i \in \{1,\ldots,N\}\). We establish the convergence result to the infimum of \(J(u)\) when \(F:\mathbb{R}\to\mathbb{R}\) is a locally Lipschitz function of controlled growth, following the Galerkin method. As an application, we establish the existence of solutions to a class of Dirichlet inclusions associated to the functional.
Keywords: anisotropic Sobolev space, Clarke generalized gradient, Dirichlet problem, Galerkin basis, \((\vec{p},\vec{q})\)-Laplacian with variable exponents.
Mathematics Subject Classification: 46E30, 47J22.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 101-119, https://doi.org/10.7494/OpMath.202511231.

 

]]> For \(0 \lt p \leq 1 \lt q \lt \infty\) and \(\gamma \gt 0\), we introduce the Calderón-Hardy spaces \(\mathcal{H}^{p}_{q,\gamma}(\mathbb{H}^{n})\) on the Heisenberg group \(\mathbb{H}^{n}\), and show for every \(f \in H^{p}(\mathbb{H}^{n})\) that the equation \[\mathcal{L}F=f\] has a unique solution \(F\) in \(\mathcal{H}^{p}_{q,2}(\mathbb{H}^{n})\), where \(\mathcal{L}\) is the sub-Laplacian on \(\mathbb{H}^{n}\), \[1 \lt q \lt \frac{n+1}{n} \quad \text{and} \quad (2n+2)\left(2+\frac{2n+2}{q}\right)^{-1} \lt p \leq 1.\] https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4604.pdf Calderón-Hardy type spaces and the Heisenberg sub-Laplacian Pablo Rocha Calderón-Hardy type spaces, Hardy type spaces, atomic decomposition, Heisenberg group, sub-Laplacian doi:10.7494/OpMath.202512221 Opuscula Math. 46, no. 1 (2026), 73-99, https://doi.org/10.7494/OpMath.202512221 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202512221 https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4604.pdf 46 1 73 99
Title: Calderón-Hardy type spaces and the Heisenberg sub-Laplacian.

Author(s): Pablo Rocha.

Abstract: For \(0 \lt p \leq 1 \lt q \lt \infty\) and \(\gamma \gt 0\), we introduce the Calderón-Hardy spaces \(\mathcal{H}^{p}_{q,\gamma}(\mathbb{H}^{n})\) on the Heisenberg group \(\mathbb{H}^{n}\), and show for every \(f \in H^{p}(\mathbb{H}^{n})\) that the equation \[\mathcal{L}F=f\] has a unique solution \(F\) in \(\mathcal{H}^{p}_{q,2}(\mathbb{H}^{n})\), where \(\mathcal{L}\) is the sub-Laplacian on \(\mathbb{H}^{n}\), \[1 \lt q \lt \frac{n+1}{n} \quad \text{and} \quad (2n+2)\left(2+\frac{2n+2}{q}\right)^{-1} \lt p \leq 1.\]
Keywords: Calderón-Hardy type spaces, Hardy type spaces, atomic decomposition, Heisenberg group, sub-Laplacian.
Mathematics Subject Classification: 42B25, 42B30, 42B35, 43A80.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 73-99, https://doi.org/10.7494/OpMath.202512221.

 

]]> In this short note we study a Harnack inequality for \(k\)-Hessian equations that involve nonlinear lower-order terms which depend on the solution and its gradient. https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4603.pdf A short note on Harnack inequality for k-Hessian equations with nonlinear gradient terms Ahmed Mohammed Giovanni Porru \(k\)-Hessian equation, \(k\)-convex solutions, Harnack inequality, Liouville property doi:10.7494/OpMath.202512261 Opuscula Math. 46, no. 1 (2026), 55-72, https://doi.org/10.7494/OpMath.202512261 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202512261 https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4603.pdf 46 1 55 72
Title: A short note on Harnack inequality for k-Hessian equations with nonlinear gradient terms.

Author(s): Ahmed Mohammed, Giovanni Porru.

Abstract: In this short note we study a Harnack inequality for \(k\)-Hessian equations that involve nonlinear lower-order terms which depend on the solution and its gradient.
Keywords: \(k\)-Hessian equation, \(k\)-convex solutions, Harnack inequality, Liouville property.
Mathematics Subject Classification: 35J60, 35J70, 35B45, 35B53.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 55-72, https://doi.org/10.7494/OpMath.202512261.

 

]]> This study investigated the asymptotic behavior of linear differential systems incorporating a proportional derivative-type (PD) differential operator. Building on the classical asymptotic convergence property of Wintner, a generalized Wintner-type asymptotic result was established for such systems. The proposed framework encompasses a wide class of time-varying coefficient matrices and extends classical asymptotic theory to equations governed by PD operators. An illustrative example is presented to demonstrate the applicability of the proposed theorem. https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4602.pdf Wintner-type asymptotic behavior of linear differential systems with a proportional derivative controller Kazuki Ishibashi Wintner-type, asymptotic behavior, linear differential systems, proportional-derivative controller doi:10.7494/OpMath.202512101 Opuscula Math. 46, no. 1 (2026), 41-54, https://doi.org/10.7494/OpMath.202512101 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202512101 https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4602.pdf 46 1 41 54
Title: Wintner-type asymptotic behavior of linear differential systems with a proportional derivative controller.

Author(s): Kazuki Ishibashi.

Abstract: This study investigated the asymptotic behavior of linear differential systems incorporating a proportional derivative-type (PD) differential operator. Building on the classical asymptotic convergence property of Wintner, a generalized Wintner-type asymptotic result was established for such systems. The proposed framework encompasses a wide class of time-varying coefficient matrices and extends classical asymptotic theory to equations governed by PD operators. An illustrative example is presented to demonstrate the applicability of the proposed theorem.
Keywords: Wintner-type, asymptotic behavior, linear differential systems, proportional-derivative controller.
Mathematics Subject Classification: 34D05, 26A24.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 41-54, https://doi.org/10.7494/OpMath.202512101.

 

]]> The analytic and formal solutions to a family of singularly perturbed partial differential equations in the complex domain involving two complex time variables are considered. The analytic continuation properties of the solution of an auxiliary problem in the Borel plane overcomes the absence of adequate domains which would guarantee summability of the formal solution. Moreover, several exponential decay rates of the difference of analytic solutions with respect to the perturbation parameter at the origin are observed, leading to several asymptotic levels relating the analytic and the formal solution. https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4601.pdf Parametric formal Gevrey asymptotic expansions in two complex time variable problems Guoting Chen Alberto Lastra Stéphane Malek singularly perturbed, formal solution, several complex variables, Cauchy problem doi:10.7494/OpMath.202512271 Opuscula Math. 46, no. 1 (2026), 5-40, https://doi.org/10.7494/OpMath.202512271 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202512271 https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4601.pdf 46 1 5 40
Title: Parametric formal Gevrey asymptotic expansions in two complex time variable problems.

Author(s): Guoting Chen, Alberto Lastra, Stéphane Malek.

Abstract: The analytic and formal solutions to a family of singularly perturbed partial differential equations in the complex domain involving two complex time variables are considered. The analytic continuation properties of the solution of an auxiliary problem in the Borel plane overcomes the absence of adequate domains which would guarantee summability of the formal solution. Moreover, several exponential decay rates of the difference of analytic solutions with respect to the perturbation parameter at the origin are observed, leading to several asymptotic levels relating the analytic and the formal solution.
Keywords: singularly perturbed, formal solution, several complex variables, Cauchy problem.
Mathematics Subject Classification: 35C10, 35R10, 35C15, 35C20.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 5-40, https://doi.org/10.7494/OpMath.202512271.

 

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