80,933 research outputs found
The existence of countably many positive solutions for nonlinear singular m-point boundary value problems on the half-line
AbstractIn this paper, by introducing a new operator, improving and generating a p-Laplace operator for some p>1, we study the existence of countably many positive solutions for nonlinear boundary value problems on the half-line (φ(u′))′+a(t)f(u(t))=0,0<t<+∞,u(0)=∑i=1m−2αiu(ξi),u′(∞)=0, where φ:R→R is the increasing homeomorphism and positive homomorphism and φ(0)=0. We show the sufficient conditions for the existence of countably many positive solutions by using the fixed-point index theory and a new fixed-point theorem in cones
Blow-up analysis of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures
This paper is concerned with the compactness of metrics of the disk with
prescribed Gaussian and geodesic curvatures. We consider a blowing-up sequence
of metrics and give a precise description of its asymptotic behavior. In
particular, the metrics blow-up at a unique point on the boundary and we are
able to give necessary conditions on its location. It turns out that such
conditions depend locally on the Gaussian curvatures but they depend on the
geodesic curvatures in a nonlocal way. This is a novelty with respect to the
classical Nirenberg problem where the blow-up conditions are local, and this
new aspect is driven by the boundary condition.Comment: 31 pages, 1 figur
Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source
This paper deals with the long-time behavior of solutions of nonlinear
reaction-diffusion equations describing formation of morphogen gradients, the
concentration fields of molecules acting as spatial regulators of cell
differentiation in developing tissues. For the considered class of models, we
establish existence of a new type of ultra-singular self-similar solutions.
These solutions arise as limits of the solutions of the initial value problem
with zero initial data and infinitely strong source at the boundary. We prove
existence and uniqueness of such solutions in the suitable weighted energy
spaces. Moreover, we prove that the obtained self-similar solutions are the
long-time limits of the solutions of the initial value problem with zero
initial data and a time-independent boundary source
On Weyl-Titchmarsh Theory for Singular Finite Difference Hamiltonian Systems
We develop the basic theory of matrix-valued Weyl-Titchmarsh M-functions and
the associated Green's matrices for whole-line and half-line self-adjoint
Hamiltonian finite difference systems with separated boundary conditions.Comment: 30 pages, to appear in J. Comput. Appl. Mat
- …