Positive solutions for a Riemann-Liouville fractional system with ϱ-Laplacian operators

This paper is concerned with the nonlinear boundary value problem (1) $\beta u''-u'+f(u)=0$, (2) $u'(0)-au(0)=0,u'(1)=0$, where $f(u)=b(c-u)\exp(-k/(1+u))$ and $\beta,a,b,c,k$ are constants. First a formal singular perturbation procedure is applied to reveal the possibility of multiple solutions of (1) and (2). Then an iteration procedure is introduced which yields sequences converging to the maximal solution from above and the minimal solution from below. A criterion for a unique solution of (1), (2) is given. It is mentioned that for certain values of the parameters multiple solutions have been found numerically. Finally, the stability of solutions of (1), (2) is discussed for certain values of the parameters. A solution $u(x)$ of (1), (2) is said to be stable if the first eigenvalue $\sigma$ of the variational equations $(1)' \beta v''-v'+[\sigma\beta+f'(u)]v=0$ and $(2)' v'(0)-av(0)=0, v'(1)=0$, is positive.

Morse index and symmetry-breaking for positive solutions of one-dimensional Hénon type equations

Global Navigation Satellite System (GNSS) provides all-weather, precise Position, Velocity and Timing (PVT) solutions from any point on the surface or near to the surface of the earth. Accuracy and global availability of solutions are the major important attributes for popularization of the system. Currently, various global and regional system (GPS, GLONASS, Galileo, Beidou, NavIC and QZSS) are in operation. Most of users of GNSS are concerned about the quality of solution obtained from the receivers including the researchers. In some published literature the precision parameters have been obtained from position solution in UTM coordinate. In this paper an attempt has been made to briefly review such efforts and discuss about the parameters used for such studies for geodetic coordinate (φ, λ, h) system because, most generic and geodetic high cost receivers provide this type of position solution. A new effort is taken up to calculate the precision parameters of GNSS based position solution in geodetic coordinate reviewing the previously reported literature. The formulas are reviewed and implemented using MATLAB to develop and utility to calculate precision parameters of satellite-based position solution. A new regional navigation satellite system IRNSS/NavIC, developed by ISRO, India has been deployed. IRNSS/ NavIC data has been analyzed using the developed utility as a case study to observe the effectiveness of such parameters. This discussion and the utility would be useful for understanding the process of GNSS data analysis for studies related to accuracy and precision.

Positive solutions for ratio-dependent predator–prey interaction systems

Solutions of some quadratic Diophantine equations

On the effect of space dimension and potential on the multiplicity of positive and nodal solutions for Kirchhoff equations

In this paper we consider a Lotka–Volterra prey–predator model with cross-diffusion of fractional type. The main purpose is to discuss the existence and nonexistence of positive steady state solutions of such a model. Here a positive solution corresponds to a coexistence state of the model. Firstly we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system. Secondly we derive some necessary conditions to ensure the existence of positive solutions, which demonstrate that if the intrinsic growth rate of the prey is too small or the death rate (or the birth rate) of the predator is too large, the model does not possess positive solutions. Thirdly we study the sufficient conditions to ensure the existence of positive solutions by using degree theory. Finally we characterize the stable/unstable regions of semi-trivial solutions and coexistence regions in parameter plane.

Chapter 6 Positive solutions for Lotka-Volterra systems with cross-diffusion

Properties of positive solutions to an elliptic equation with negative exponent

Classification and refined singularity of positive solutions for nonlinear Maxwell equations arising in mesoscopic electromagnetism

Asymptotic Behavior of Solutions for the Chafee-Infante Equation

Periodic, permanent, and extinct solutions to population models

We deal with the existence and localization of positive radial solutions for Dirichlet problems involving ‐Laplacian operators in a ball. In particular, ‐Laplacian and Minkowski‐curvature equations are considered. Our approach relies on fixed point index techniques, which work thanks to a Harnack‐type inequality in terms of a seminorm. As a consequence of the localization result, it is also derived the existence of several (even infinitely many) positive solutions.

This paper considers the dynamics of the following chemotaxis system $$ \begin{cases} u_t=\Delta u-\chi\nabla (u\cdot \nabla v)+u\left(a_0(t,x)-a_1(t,x)u-a_2(t,x)\int_{\Omega}u\right),\quad x\in \Omega\cr 0=\Delta v+ u-v,\quad x\in \Omega \quad \cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,\quad x\in\partial\Omega, \end{cases} $$ where $\Omega \subset \mathbb{R}^n(n\geq 1)$ is a bounded domain with smooth boundary $\partial\Omega$ and $a_i(t,x)$ ($i=0,1,2$) are locally H\"older continuous in $t\in\mathbb{R}$ uniformly with respect to $x\in\bar{\Omega}$ and continuous in $x\in\bar{\Omega}$. We first prove the local existence and uniqueness of classical solutions $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with $u(x,t_0;t_0,u_0)=u_0(x)$ for various initial functions $u_0(x)$. Next, under some conditions on the coefficients $a_1(t,x)$, $a_2(t,x)$, $\chi$ and $n$, we prove the global existence and boundedness of classical solutions $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with given nonnegative initial function $u(x,t_0;t_0,u_0)=u_0(x)$. Then, under the same conditions for the global existence, we show that the system has an entire positive classical solution $(u^*(x,t),v^*(x,t))$. Moreover, if $a_i(t,x)$ $(i=0,1,2)$ are periodic in $t$ with period $T$ or are independent of $t$, then the system has a time periodic positive solution $(u^*(x,t),v^*(x,t))$ with periodic $T$ or a steady state positive solution $(u^*(x),v^*(x))$. If $a_i(t,x)$ $(i=0,1,2)$ are independent of $x$ , then the system has a spatially homogeneous entire positive solution $(u^*(t),v^*(t))$. Finally, under some further assumptions, we prove that the system has a unique entire positive solution $(u^*(x,t),v^*(x,t))$ which is globally stable . Moreover, if $a_i(t,x)$ $(i=0,1,2)$ are periodic or almost periodic in $t$, then $(u^*(x,t),v^*(x,t))$ is also periodic or almost periodic in $t$.

The first two authors [Proc. Lond. Math. Soc. (3) {\bf 114}(1):1--34, 2017] classified the behaviour near zero for all positive solutions of the perturbed elliptic equation with a critical Hardy--Sobolev growth $$-\Delta u=|x|^{-s} u^{2^\star(s)-1} -\mu u^q \hbox{ in }B\setminus\{0\},$$ where $B$ denotes the open unit ball centred at $0$ in $\mathbb{R}^n$ for $n\geq 3$, $s\in (0,2)$, $2^\star(s):=2(n-s)/(n-2)$, $\mu>0$ and $q>1$. For $q\in (1,2^\star-1)$ with $2^\star=2n/(n-2)$, it was shown in the op. cit. that the positive solutions with a non-removable singularity at $0$ could exhibit up to three different singular profiles, although their existence was left open. In the present paper, we settle this question for all three singular profiles in the maximal possible range. As an important novelty for $\mu>0$, we prove that for every $q\in (2^\star(s) -1,2^\star-1)$ there exist infinitely many positive solutions satisfying $|x|^{s/(q-2^\star(s)+1)}u(x)\to \mu^{-1/(q-2^\star(s)+1)}$ as $|x|\to 0$, using a dynamical system approach. Moreover, we show that there exists a positive singular solution with $\liminf_{|x|\to 0} |x|^{(n-2)/2} u(x)=0$ and $\limsup_{|x|\to 0} |x|^{(n-2)/2} u(x)\in (0,\infty)$ if (and only if) $q\in (2^\star-2,2^\star-1)$.