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Opuscula Mathematica
Scopus
ISSN 1232-9274 · AGH Univeristy of Science and Technology Press · 943 artikel di indeks
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Galerkin-type minimizers to a competing problem for (\vec{p},\vec{q})-Laplacian with variable exponents
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…
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Parametric formal Gevrey asymptotic expansions in two complex time variable problems
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…
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Calderón-Hardy type spaces and the Heisenberg sub-Laplacian
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…
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A short note on Harnack inequality for k-Hessian equations with nonlinear gradient terms
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.
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Wintner-type asymptotic behavior of linear differential systems with a proportional derivative controller
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…
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Ground states for the planar indefinite Choquard equation with supercritical exponential growth
This paper investigates a class of strongly indefinite Choquard equations formulated on the two-dimensional unit ball \(B\). Specifically, consider the following equation: \[\begin{cases}-\Delta u+V(x)u=(I_{\mu}\ast F(u))f(u), &x\in B, \\ u\in H^1_{0,\mathrm{rad}}(B), & \end{cases}\] where…
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Normalized solutions for p-Kirchhoff equation with general nonlinearities on bounded domain
In this paper, we study the existence of the solutions for the following \(p\)-Kirchhoff equation on bounded domain \[\begin{cases}-\Bigg(a+b\displaystyle\int_{\Omega}|\nabla u|^pdx\Bigg)\Delta_pu+\lambda|u|^{p-2}u=h(u), &x \in \Omega,\\ u=0, &x\in\partial\Omega,\end{cases}\] with prescribed mass…
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A note on the development of singularities on solutions to the Navier-Stokes equations under supercritical forcing terms
Recently, Qi S. Zhang provided examples of solutions to the Navier-Stokes equations which, under suitable hypotheses, blow-up in finite time. He considers axially symmetric solutions in a cylinder \(D\) under appropriate boundary conditions and under the effect of supercritical external forces…
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Nonlinear hybrid second-order neutral delay differential equation with mixed sign terms: Oscillation via symmetry transform
This paper investigates the oscillatory behavior of solutions to a class of second-order nonlinear neutral delay differential equations with both positive and negative terms of the form \[\left(a(\theta) z^{\prime}(\theta)\right)^{\prime} - p(\theta)x(\theta) + q(\theta)x^{\alpha}(\sigma(\theta)) =…
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One complex variable versus several complex variables: The Hartogs extension phenomenon in context
We explicate the difference between the function theory of one complex variable and the function theory of several complex variables. In particular, we use the inhomogeneous Cauchy-Riemann equations to explain why there is a Hartogs extension phenomenon in several complex variables but not in one…
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Topological sensitivity analysis for the narrow escape problem
The topological sensitivity analysis method has been recognized as a promising, fast, and accurate approach for solving topology optimization and inverse problems. It is based on developing an asymptotic expansion of a design functional with respect to the creation of a small hole inside the…
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An elliptic system involving p-Laplacian and a chemotactic term
We study the nonlinear elliptic system: \[\begin{cases} u \in W^{1,p}_0(\Omega): -\operatorname{div} (a(x)|\nabla u|^{p-2}\nabla u) + u = -\operatorname{div} (a(x) u|\nabla \psi|^{p-2}\nabla \psi) + f(x), \\ \psi \in W^{1,p}_0(\Omega): -\operatorname{div} (a(x)|\nabla \psi |^{p-2}\nabla \psi) =…
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Solutions with prescribed mass for a critical Choquard equation driven by a local-nonlocal operator
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,\\…
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Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains
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…
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Minimum k-critical-bipartite graphs: the irregular case
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…
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On the existence of independent (1,k)-dominating sets for k\in\{1,2\} in two products of graphs
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…
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On a fixed point theorem for operator systems and eigenvalue criteria for existence of positive solutions
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…
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Damped nonlinear Ginzburg-Landau equation with saturation. Part II. Strong Stabilization
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…
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Artikel Jurnal Teks lengkap
Some remarks and results on the Standard (2,2)-Conjecture
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…
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Normalized ground states for a p-Laplacian system in the mass super-critical case
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…