Opuscula Mathematica

Let $s\in (0,1)$ and $N\gt 2s$. In this paper, we consider the following class of nonlocal semipositone problems: $(-\Delta)^s u= g(x)f_a(u)\text{ in }\mathbb{R}^N,\quad u \gt 0\text{ in }\mathbb{R}^N,$ where the weight $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is positive, $a\gt 0$ is a parameter, and $f_a \in \mathcal{C}(\mathbb{R})$ is strictly negative on $(-\infty,0]$. For $f_a$ having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution $u_a$, provided a is near zero. To obtain the positivity of $u_a$, we establish a Brezis-Kato type uniform estimate of $(u_a)$ in $L^r(\mathbb{R}^N)$ for every $r \in [\frac{2N}{N-2s}, \infty]$.

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Degenerate singular parabolic problems with natural growth

(Wydawnictwa AGH, 2024) El Ouardy, Mounim; El Hadfi, Youssef; Sbai, Abdelaaziz

In this paper, we study the existence and regularity results for nonlinear singular parabolic problems with a natural growth gradient term $\$ \begin{cases}\frac{\partial u}{\partial t}-\operatorname{div}((a(x,t)+u^{q})|\nabla u|^{p-2}\nabla u)+d(x,t)\frac{|\nabla u|^{p}}{u^{\gamma}}=f & \text{ in } Q,\\ u(x,t)=0 & \text{ on } \Gamma, \\ u(x,t=0)=u_{0}(x) & \text{ in } \Omega, \end{cases} \$$ where $\Omega$ is a bounded open subset of $\mathbb{R}^{N}$, $N\gt 2$, $Q$ is the cylinder $\Omega \times (0,T)$, $T\gt 0$, $\Gamma$ the lateral surface $\partial \Omega \times (0,T)$, $2\leq p\lt N$, $a(x,t)$ and $b(x,t)$ are positive measurable bounded functions, $q\geq 0$, $0\leq\gamma\lt 1$, and $f$ non-negative function belongs to the Lebesgue space $L^{m}(Q)$ with $m\gt 1$, and $u_{0}\in L^{\infty}(\Omega)$ such that $\$\forall\omega\subset\subset\Omega\, \exists D_{\omega}\gt 0:\, u_{0}\geq D_{\omega}\text{ in }\omega.\$$ More precisely, we study the interaction between the term $u^{q}$ $(q>0)$ and the singular lower order term $d(x,t)|\nabla u|^{p}u^{-\gamma}$ $(0\lt\gamma\lt 1)$ in order to get a solution to the above problem. The regularizing effect of the term $u^q$ on the regularity of the solution and its gradient is also analyzed.

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Asymptotic analysis for confluent hypergeometric function in two variables given by double integral

(Wydawnictwa AGH, 2024) Haraoka, Yoshishige

We study an integrable connection with irregular singularities along a normally crossing divisor. The connection is obtained from an integrable connection of regular singular type by a confluence, and has irregular singularities along $x=\infty$ and $y=\infty$. Solutions are expressed by a double integral of Euler type with resonances among the exponents in the integrand. We specify twisted cycles that give main asymptotic behaviors in sectorial domains around $(\infty,\infty)$. Then we obtain linear relations among the twisted cycles, and get an explicit expression of the Stokes multiplier. The methods to derive the asymptotic behaviors for double integrals and to get linear relations among twisted cycles in resonant case, which we developed in this paper, seem to be new.We study an integrable connection with irregular singularities along a normally crossing divisor. The connection is obtained from an integrable connection of regular singular type by a confluence, and has irregular singularities along $x=\infty$ and $y=\infty$. Solutions are expressed by a double integral of Euler type with resonances among the exponents in the integrand. We specify twisted cycles that give main asymptotic behaviors in sectorial domains around $(\infty,\infty)$. Then we obtain linear relations among the twisted cycles, and get an explicit expression of the Stokes multiplier. The methods to derive the asymptotic behaviors for double integrals and to get linear relations among twisted cycles in resonant case, which we developed in this paper, seem to be new.

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Graphs whose vertex set can be partitioned into a total dominating set and an independent dominating set

(Wydawnictwa AGH, 2024) Haynes, Teresa W.; Henning, Michael A.

A graph $G$ whose vertex set can be partitioned into a total dominating set and an independent dominating set is called a TI-graph. We give constructions that yield infinite families of graphs that are TI-graphs, as well as constructions that yield infinite families of graphs that are not TI-graphs. We study regular graphs that are TI-graphs. Among other results, we prove that all toroidal graphs are TI-graphs.

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Geometric properties of the lattice of polynomials with integer coefficients

(Wydawnictwa AGH, 2024) Lipnicki, Artur; Śmietański, Marek J.

This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let $r$, $n$ be positive integers with $n \ge 6r$. Let $\boldsymbol{P}_n \cap \boldsymbol{M}_r$ be the space of polynomials of degree at most $n$ on $[0,1]$ with integer coefficients such that $P^{(k)}(0)/k!$ and $P^{(k)}(1)/k!$ are integers for $k=0,\dots,r-1$ and let $\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r$ be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of $\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r$ from $\boldsymbol{P}_n \cap \boldsymbol{M}_r$ in $L_2(0,1)$. We give rather precise quantitative estimations for successive minima of $\boldsymbol{P}_n^\mathbb{Z}$ in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in $\boldsymbol{P}_n \cap \boldsymbol{M}_r$.

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On the solvability of some parabolic equations involving nonlinear boundary conditions with L1 data

(Wydawnictwa AGH, 2024) Taourirte, Laila; Charkaoui, Abderrahim; Alaa, Nour Eddine

We analyze the existence of solutions for a class of quasilinear parabolic equations with critical growth nonlinearities, nonlinear boundary conditions, and $L^1$ data. We formulate our problems in an abstract form, then using some techniques of functional analysis, such as Leray-Schauder's topological degree associated with the truncation method and very interesting compactness results, we establish the existence of weak solutions to the proposed models.

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