Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2015
Volume
Vol. 35
Number
No. 1
Description
Journal Volume
Opuscula Mathematica
Vol. 35 (2015)
Projects
Pages
Articles
Positive solutions with specific asymptotic behavior for a polyharmonic problem on Rn
(2015) Dhifli, Abdelwaheb
This paper is concerned with positive solutions of the semilinear polyharmonic equation $(-\Delta)^{m} u = a(x){u}^{\alpha}$ on $\mathbb{R}^{n}$, where $m$ and $n$ are positive integers with $n\gt 2m$, $\alpha\in (-1,1)$. The coefficient a is assumed to satisfy $a(x)\approx{(1+|x|)}^{-\lambda}L(1+|x|)\quad \text{for}\quad x\in \mathbb{R}^{n},$ where $\lambda\in [2m,\infty)$ and $L\in C^{1}([1,\infty))$ is positive with $\frac{tL'(t)}{L(t)}\longrightarrow 0$ as $t\longrightarrow \infty$; if $\lambda=2m$, one also assumes that $\int_{1}^{\infty}t^{-1}L(t)dt\lt \infty$. We prove the existence of a positive solution $u$ such that $u(x)\approx{(1+|x|)}^{-\widetilde{\lambda}}\widetilde{L}(1+|x|) \quad\text{for}\quad x\in \mathbb{R}^{n},$ with $\widetilde{\lambda}:=\min(n-2m,\frac{\lambda-2m}{1-\alpha})$ and a function $\widetilde{L}$, given explicitly in terms of $L$ and satisfying the same condition at infinity. (Given positive functions $f$ and $g$ on $\mathbb{R}^{n}$, $f\approx g$ means that $c^{-1}g\leq f\leq cg$ for some constant $c\gt 1$.)
Asymptotic behavior of positive solutions of a semilinear Dirichlet problem in the annulus
(2015) Dridi, Safa; Khamessi, Bilel
In this paper, we establish existence and asymptotic behavior of a positive classical solution to the following semilinear boundary value problem: $-\Delta u=q(x)u^{\sigma }\;\text{in}\;\Omega,\quad u_{|\partial\Omega}=0.$. Here $\Omega$ is an annulus in $\mathbb{R}^{n}$, $n\geq 3$, $\sigma \lt 1$ and $q$ is a positive function in $\mathcal{C}_{loc}^{\gamma }(\Omega )$, $0\lt\gamma \lt 1$, satisfying some appropriate assumptions related to Karamata regular variation theory. Our arguments combine a method of sub- and supersolutions with Karamata regular variation theory
Characterizations and decomposition of strongly Wright-convex functions of higher order
(2015) Gilányi, Attila; Merentes, Nelson; Nikodem, Kazimierz; Páles, Zsolt
Motivated by results on strongly convex and strongly Jensen-convex functions by R. Ger and K. Nikodem in [Strongly convex functions of higher order, Nonlinear Anal. 74 (2011), 661–665] we investigate strongly Wright-convex functions of higher order and we prove decomposition and characterization theorems for them. Our decomposition theorem states that a function $f$ is strongly Wright-convex of order $n$ if and only if it is of the form $f(x)=g(x)+p(x)+c x^{n+1}$, where $g$ is a (continuous) n-convex function and $p$ is a polynomial function of degree $n$. This is a counterpart of Ng’s decomposition theorem for Wright-convex functions. We also characterize higher order strongly Wright-convex functions via generalized derivatives.
Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities
(2015) Jaroš, Jaroslav; Takashi, Kusano
We consider n-dimensional cyclic systems of second order differential equations $(p_i(t)|x_{i}'|^{\alpha_i -1}x_{i}')' = q_{i}(t)|x_{i+1}|^{\beta_i-1}x_{i+1},$ $\quad i = 1,\ldots,n, \quad (x_{n+1} = x_1) \tag{\(\ast\)}$ $(*)$ under the assumption that the positive constants $\alpha_i$ and $\beta_i$ satisfy $\alpha_1{\ldots}\alpha_n \gt \beta_1{\ldots}\beta_n$ and $q_i(t)$ are regularly varying functions, and analyze positive strongly increasing solutions of system $(*)$ in the framework of regular variation. We show that the situation for the existence of regularly varying solutions of positive indices for $(*)$ can be characterized completely, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of growth precisely. We give examples demonstrating that the main results for $(*)$ can be applied to some classes of partial differential equations with radial symmetry to acquire accurate information about the existence and the asymptotic behavior of their radial positive strongly increasing solutions.
Controllability of semilinear systems with fixed delay in control
(2015) Kumar, Surendra; Sukavanam, N.
In this paper, different sufficient conditions for exact controllability of semilinear systems with a single constant point delay in control are established in infinite dimensional space. The existence and uniqueness of mild solution is also proved under suitable assumptions. In particular, local Lipschitz continuity of a nonlinear function is used. To illustrate the developed theory some examples are given.