Browsing by Subject "positive solutions"

Now showing 1 - 10 of 10

Access status: Open Access ,
Existence and boundary behavior of positive solutions for a Sturm-Liouville problem
(2016) Masmoudi, Syrine; Zermani, Samia
In this paper, we discuss existence, uniqueness and boundary behavior of a positive solution to the following nonlinear Sturm-Liouville problem $\begin{aligned}&\frac{1}{A}(Au^{\prime })^{\prime }+a(t)u^{\sigma}=0\;\;\text{in}\;(0,1),\\ &\lim\limits_{t\to 0}Au^{\prime}(t)=0,\quad u(1)=0,\end{aligned}$ where $\sigma \lt 1$, $A$ is a positive differentiable function on $(0,1)$ and $a$ is a positive measurable function in $(0,1)$ satisfying some appropriate assumptions related to the Karamata class. Our main result is obtained by means of fixed point methods combined with Karamata regular variation theory.
Access status: Open Access ,
Existence of positive solutions to a discrete fractional boundary value problem and corresponding Lyapunov-type inequalities
(Wydawnictwa AGH, 2018) Chidouh, Amar; Torres, Delfim F. M.
We prove existence of positive solutions to a boundary value problem depending on discrete fractional operators. Then, corresponding discrete fractional Lyapunov-type inequalities are obtained.
Access status: Open Access ,
Further properties of the rational recursive sequence xn + 1 = axn - 1 / (b + cxnxn - 1)
(2006) Andruch-Sobiło, Anna; Migda, Małgorzata
In this paper we consider the difference equation $x_{n+1}=\frac{ax_{n-1}}{b+cx_{n}x_{n-1}}, \quad n=0,1,...(E)$ with positive parameters $a$ and $c$, negative parameter $b$ and nonnegative initial conditions. We investigate the asymptotic behavior of solutions of equation $\text{(E)}$.
Access status: Open Access ,
On a Robin (p,q)-equation with a logistic reaction
(Wydawnictwa AGH, 2019) Papageorgiou, Nikolaos Socrates; Vetro, Calogero; Vetro, Francesca
We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a $p$-Laplacian and of a $q$-Laplacian ($(p,q)$-equation) plus an indefinite potential term and a parametric reaction of logistic type (superdiffusive case). We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $\lambda \gt 0$ varies. Also, we show that for every admissible parameter $\lambda \gt 0$, the problem admits a smallest positive solution.
Access status: Open Access ,
Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions
(Wydawnictwa AGH, 2019) Hai, D. D.; Wang, X.
We prove the existence of positive solutions for the $p$-Laplacian problem $\begin{cases}-(r(t)\phi (u^{\prime }))^{\prime }=\lambda g(t)f(u),& t\in (0,1),\\au(0)-H_{1}(u^{\prime }(0))=0,\\cu(1)+H_{2}(u^{\prime}(1))=0,\end{cases}$ where $\phi (s)=|s|^{p-2}s$, $p \gt 1$, $H_{i}:\mathbb{R}\rightarrow\mathbb{R}$ can be nonlinear, $i=1,2$, $f:(0,\infty)\rightarrow \mathbb{R}$ is $p$-superlinear or $p$-sublinear at $\infty$ and is allowed be singular $(\pm\infty)$ at $0$, and $\lambda$ is a positive parameter.
Access status: Open Access ,
Positive solutions of boundary value problems with nonlinear nonlocal boundary conditions
(2016) Padhi, Seshadev; Pati, Smita; Hota, D. K.
We consider the existence of at least three positive solutions of a nonlinear first order problem with a nonlinear nonlocal boundary condition given by $\begin{aligned} x^{\prime}(t)& = r(t)x(t) + \sum_{i=1}^{m} f_i(t,x(t)), \quad t \in [0,1],\\ \lambda x(0)& = x(1) + \sum_{j=1}^{n} \Lambda_j(\tau_j, x(\tau_j)),\quad \tau_j \in [0,1],\end{aligned}$ where $r:[0,1] \rightarrow [0,\infty)$ is continuous; the nonlocal points satisfy $0 \leq \tau_1 \lt \tau_2 \lt \ldots \lt \tau_n \leq 1$ the nonlinear function $f_i$ and $\tau_j$ are continuous mappings from $[0,1] \times [0,\infty) \rightarrow [0,\infty)$ for $i = 1,2,\ldots ,m$ and $j = 1,2,\ldots ,n$ respectively, and $\lambda \gt 0$ is a positive parameter.
Access status: Open Access ,
Positive solutions of nonpositone sublinear elliptic problems
(Wydawnictwa AGH, 2024) Godoy, Tomas
Consider the problem $-\Delta u=\lambda f(\cdot, u) $ in $\Omega$, $u=0$ on $\partial\Omega$, $u\gt 0$ in $\Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $C^{2}$ boundary when $n\geq2$, $\lambda\gt 0$, and where $f\in C (\overline{\Omega}\times[0,\infty)) $ satisfies $\lim_{s\rightarrow\infty}s^{-p}f(\cdot, s) =\gamma$ for some $p\in(0,1)$ and some $\gamma\in C(\overline{\Omega}) $ such that $\gamma\neq 0$ a.e. in $\Omega$ and, for some positive constants $c$ and $c^{\prime}$, $\gamma^{-}\leq cd_{\Omega}^{\beta}$ for some $\beta\in (\frac{n-1}{n},\infty)$ and $(-\Delta)^{-1}\gamma\geq c^{\prime}d_{\Omega}$, where $d_{\Omega}(x):=dist ( x,\partial \Omega) $ and $\gamma^{-}:=-\min(0,\gamma)$. Under these assumptions we show that for $\lambda$ large enough, the above problem has a positive weak solution $u\in C^{1}(\overline{\Omega})$ such that, for some constant $c^{\prime\prime}\gt 0$, $u\geq c^{\prime\prime}d_{\Omega}$ in $\Omega$.
Access status: Open Access ,
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.
Access status: Open Access ,
Study of fractional semipositone problems on RN
(Wydawnictwa AGH, 2024) Biswas, Nirjan
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 $r \in [\frac{2N}{N-2s}, \infty]$.
Access status: Open Access ,
Uniqueness for a class p-Laplacian problems when a parameter is large
(Wydawnictwa AGH, 2024) Alreshidi, Bandar; Hai, D. D.
We prove uniqueness of positive solutions for the problem $-\Delta_{p}u=\lambda f(u)\text{ in }\Omega,\ u=0\text{ on }\partial \Omega,$ where $1\lt p\lt 2$ and $p$ is close to 2, $\Omega$ is bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial \Omega$, $f:[0,\infty)\rightarrow [0,\infty )$ with $f(z)\sim z^{\beta }$ at $\infty$ for some $\beta \in (0,1)$, and $\lambda$ is a large parameter. The monotonicity assumption on $f$ is not required even for u large.