25 research outputs found
Positive Solutions for Fractional p- Laplace Semipositone Problem with Superlinear Growth
We consider a semipositone problem involving the fractional Laplace
operator of the form \begin{equation*} \begin{aligned} (-\Delta)_p^s u &=\mu(
u^{r}-1) \text{ in } \Omega,\\ u &>0 \text{ in }\Omega,\\ u &=0 \text{ on
}\Omega^{c}, \end{aligned} \end{equation*} where is a smooth bounded
convex domain in , , where
, and is a positive parameter. We study the
behaviour of the barrier function under the fractional -Laplacian and use
this information to prove the existence of a positive solution for small
using degree theory. Additionally, the paper explores the existence of a ground
state positive solution for a multiparameter semipositone problem with critical
growth using variational arguments.Comment: 35 pages, 0 figure
Positive solutions of nonlinear elliptic boundary value problems
This dissertation focuses on the study of positive steady states to classes of nonlinear reaction diffusion (elliptic) systems on bounded domains as well as on exterior domains with Dirichlet boundary conditions. In particular, we study such systems in the challenging case when the reaction terms are negative at the origin, referred in the literature as semipositone problems. For the last 30 years, study of elliptic partial differential equations with semipositone structure has flourished not only for the semilinear case but also for quasilinear case. Here we establish several results that directly contribute to and enhance the literature of semipositone problems. In particular, we discuss existence, non-existence and multiplicity results for classes of superlinear as well as sublinear systems. We establish our results via the method of sub-super solutions, degree theory arguments, a priori bounds and energy analysis
Positive solutions for nonlinear singular differential systems involving parameter on the half-line
By using the upper-lower solutions method and the fixed-point theorem on cone in a special space, we study the singular boundary value problem for systems of nonlinear second-order differential equations involving two parameters on the half-line. Some results for the existence, nonexistence and multiplicity of positive solutions for the problem are obtained
Nonnegative solutions of nonlinear fractional Laplacian equations
The study of reaction-diffusion equations involving nonlocal diffusion operators has recently flourished. The fractional Laplacian is an example of a nonlocal diffusion operator which allows long-range interactions in space, and it is therefore important from the application point of view. The fractional Laplacian operator plays a similar role in the study of nonlocal diffusion operators as the Laplacian operator does in the local case. Therefore, the goal of this dissertation is a systematic treatment of steady state reaction-diffusion problems involving the fractional Laplacian as the diffusion operator on a bounded domain and to investigate existence (and nonexistence) results with respect to a bifurcation parameter. In particular, we establish existence results for positive solutions depending on the behavior of a nonlinear reaction term near the origin and at infinity. We use topological degree theory as well as the method of sub- and supersolutions to prove our existence results. In addition, using a moving plane argument, we establish that, for a class of steady state reaction-diffusion problems involving the fractional Laplacian, any nonnegative nontrivial solution in a ball must be positive, and hence radially symmetric and radially decreasing. Finally, we provide numerical bifurcation diagrams and the profiles of numerical positive solutions, corresponding to theoretical results, using finite element methods in one and two dimensions
Existence results for a coupled system of nonlinear fractional differential equations with boundary value problems on an unbounded domain
This paper deals with the existence results for solutions of coupled system of nonlinear fractional differential equations with boundary value problems on an unbounded domain. Also, we give an illustrative example in order to indicate the validity of our assumptions
Solvability and asymptotic properties for an elliptic geophysical fluid flows model in a planar exterior domain
In this paper, we study the solvability and asymptotic properties of a recently derived gyre model of nonlinear elliptic Schrödinger equation arising from the geophysical fluid flows. The existence theorems and the asymptotic properties for radial positive solutions are established due to space theory and analytical techniques, some special cases and specific examples are also given to describe the applicability of model in gyres of geophysical fluid flows
Symmetry in Modeling and Analysis of Dynamic Systems
Real-world systems exhibit complex behavior, therefore novel mathematical approaches or modifications of classical ones have to be employed to precisely predict, monitor, and control complicated chaotic and stochastic processes. One of the most basic concepts that has to be taken into account while conducting research in all natural sciences is symmetry, and it is usually used to refer to an object that is invariant under some transformations including translation, reflection, rotation or scaling.The following Special Issue is dedicated to investigations of the concept of dynamical symmetry in the modelling and analysis of dynamic features occurring in various branches of science like physics, chemistry, biology, and engineering, with special emphasis on research based on the mathematical models of nonlinear partial and ordinary differential equations. Addressed topics cover theories developed and employed under the concept of invariance of the global/local behavior of the points of spacetime, including temporal/spatiotemporal symmetries