78 research outputs found
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
Elliptic problems on complete non-compact Riemannian manifolds with asymptotically non-negative Ricci curvature
In this paper we discuss the existence and non--existence of weak solutions
to parametric equations involving the Laplace-Beltrami operator in a
complete non-compact --dimensional () Riemannian manifold
with asymptotically non--negative Ricci curvature and
intrinsic metric . Namely, our simple model is the following problem
\left\{ \begin{array}{ll} -\Delta_gw+V(\sigma)w=\lambda \alpha(\sigma)f(w) &
\mbox{ in } \mathcal{M}\\ w\geq 0 & \mbox{ in } \mathcal{M} \end{array}\right.
where is a positive coercive potential, is a positive bounded
function, is a real parameter and is a suitable continuous
nonlinear term. The existence of at least two non--trivial bounded weak
solutions is established for large value of the parameter requiring
that the nonlinear term is non--trivial, continuous, superlinear at zero
and sublinear at infinity. Our approach is based on variational methods. No
assumptions on the sectional curvature, as well as symmetry theoretical
arguments, are requested in our approach
Book of Abstracts
USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud
São Carlos, Brasil. 3-7 february 2014
The Obstacle Problem at Zero for the Fractional p-Laplacian
In this paper we establish a multiplicity result for a class of unilateral, nonlinear, nonlocal problems with nonsmooth potential (variational-hemivariational inequalities), using the degree map of multivalued perturbations of fractional nonlinear operators of monotone type, the fact that the degree at a local minimizer of the corresponding Euler functional is equal one, and controlling the degree at small balls and at big balls
EQUADIFF 15
Equadiff 15 – Conference on Differential Equations and Their Applications – is an international conference in the world famous series Equadiff running since 70 years ago. This booklet contains conference materials related with the 15th Equadiff conference in the Czech and Slovak series, which was held in Brno in July 2022. It includes also a brief history of the East and West branches of Equadiff, abstracts of the plenary and invited talks, a detailed program of the conference, the list of participants, and portraits of four Czech and Slovak outstanding mathematicians
Beyond the classical strong maximum principle: forcing changing sign near the boundary and flat solutions
We show that the classical strong maximum principle, concerning positive
supersolutions of linear elliptic equations vanishing on the boundary of the
domain can be extended, under suitable conditions, to the case in
which the forcing term is changing sign. In addition, in the case of
solutions, the normal derivative on the boundary may also vanish on the
boundary (definition of flat solutions). This leads to examples in which the
unique continuation property fails. As a first application, we show the
existence of positive solutions for a sublinear semilinear elliptic problem of
indefinite sign. A second application, concerning the positivity of solutions
of the linear heat equation, for some large values of time, with forcing and/or
initial datum changing sign is also given.Comment: 20 pages 2 Figure
Applied Mathematics and Fractional Calculus
In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia
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