Decaying positive solutions of some quasilinear differential equations

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A weak comparison principle for some quasilinear elliptic operators: it compares functions belonging to different spaces

Akihito Unai (2018)

Applications of Mathematics

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We shall prove a weak comparison principle for quasilinear elliptic operators $- div (a (x, \nabla u))$ that includes the negative $p$ -Laplace operator, where $a : Ω \times ℝ^{N} \to ℝ^{N}$ satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces.

A bound for the solutions of a basic elliptic system with non-linearity $q\geq 2$

Sergio Campanato (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In questa Nota si dimostra un risultato enunciato nel § 5 della pubblicazione [4]. Per le soluzioni di un sistema ellittico base, con non-linearità $q\geq 2$ , vale un principio di massimo analogo a quello dimostrato in [3] nel caso di non-linearità $q=2$ .

On a generalized Dhombres functional equation. II.

P. Kahlig, Jaroslav Smítal (2002)

Mathematica Bohemica

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We consider the functional equation $f (x f (x)) = ϕ (f (x))$ where $ϕ J \to J$ is a given increasing homeomorphism of an open interval $J \subset (0, \infty)$ and $f (0, \infty) \to J$ is an unknown continuous function. In a previous paper we proved that no continuous solution can cross the line $y = p$ where $p$ is a fixed point of $ϕ$ , with a possible exception for $p = 1$ . The range of any non-constant continuous solution is an interval whose end-points are fixed by $ϕ$ and which contains in its interior no fixed point except for $1$ . We also gave a characterization of the...

Filling boxes densely and disjointly

J. Schröder (2003)

Commentationes Mathematicae Universitatis Carolinae

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We effectively construct in the Hilbert cube $ℍ = {[0, 1]}^{ω}$ two sets $V, W \subset ℍ$ with the following properties: (a) $V \cap W = \emptyset$ , (b) $V \cup W$ is discrete-dense, i.e. dense in ${[0, 1]}_{D}^{ω}$ , where ${[0, 1]}_{D}$ denotes the unit interval equipped with the discrete topology, (c) $V$ , $W$ are open in $ℍ$ . In fact, $V = ⋃_{ℕ} V_{i}$ , $W = ⋃_{ℕ} W_{i}$ , where $V_{i} = ⋃_{0}^{2^{i - 1} - 1} V_{i j}$ , $W_{i} = ⋃_{0}^{2^{i - 1} - 1} W_{i j}$ . $V_{i j}$ , $W_{i j}$ are basic open sets and $(0, 0, 0, ...) \in V_{i j}$ , $(1, 1, 1, ...) \in W_{i j}$ , (d) $V_{i} \cup W_{i}$ , $i \in ℕ$ is point symmetric about $(1 / 2, 1 / 2, 1 / 2, ...)$ . Instead of $[0, 1]$ we could have taken any $T_{4}$ -space or a digital interval, where the resolution (number of points) increases with $i$ .

Existence of positive radial solutions for the elliptic equations on an exterior domain

Yongxiang Li, Huanhuan Zhang (2016)

Annales Polonici Mathematici

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We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩ $l i m_{| x | \to \infty} u (x) = 0$ , where $Ω = x \in ℝ^{N} : | x | > r ₀$ , N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and $0 < \int_{r ₀}^{\infty} r K (r) d r < \infty$ , f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the...

Distributivity of strong implications over conjunctive and disjunctive uninorms

Daniel Ruiz-Aguilera, Joan Torrens (2006)

Kybernetika

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This paper deals with implications defined from disjunctive uninorms $U$ by the expression $I (x, y) = U (N (x), y)$ where $N$ is a strong negation. The main goal is to solve the functional equation derived from the distributivity condition of these implications over conjunctive and disjunctive uninorms. Special cases are considered when the conjunctive and disjunctive uninorm are a $t$ -norm or a $t$ -conorm respectively. The obtained results show a lot of new solutions generalyzing those obtained in previous works...

A bound for the solutions of a basic elliptic system with non-linearity $q\geq 2$

Sergio Campanato (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Solutions to a class of singular quasilinear elliptic equations

Lin Wei, Zuodong Yang (2010)

Annales Polonici Mathematici

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We study the existence of positive solutions to ⎧ ${d i v (| \nabla u |}^{p - 2} \nabla u) + q (x) u^{- γ} = 0$ on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is $ℝ^{N}$ or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).