80,933 research outputs found

    The existence of countably many positive solutions for nonlinear singular m-point boundary value problems on the half-line

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    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

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    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

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    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
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