Multiplicity of Solutions of the Weighted p-Laplacian with Isolated Singularity and Diffusion Suppressed by Convection

In this paper,the authors study the multiplicity of solutions to the weighted p-Laplacian with isolated singularity and diffusion suppressed by convection-div(|x|^(α)|■u|^(p-2)■u)+λ^(1)/|x|β|■u|^(p-2)■u·x=|x|^(γ)g(|x|)in B{0}subject to nonlinear Robin boundary value condition|x|^(α)|■u|^(p-2)■u·n=A-ρu on■B,whereλ>0,B■RN(N≥2)is the unit ball centered at the origin,α>0,p>1,β∈R,γ>-N,g∈C(0,1])with g(0)>0,A∈R,ρ>0 and n is the unit outward normal.The same problem with diffusion promoted by convection,namely λ≤0,has already been discussed by the last two authors(Song-Yin(2012)),where the existence,nonexistence and classification of singularities for solutions are presented.Completely different from[Song,H.J.and Yin,J.X.,Removable isolated singularities of solutions to the weighted p-Laplacian with singular convection,Math.Meth.Appl.Sci.,35,2012,1089-1100],in the present caseλ>0,namely the diffusion is suppressed by the convection,non-singular solutions are not only existent but also may be infinite which vary according only to the values of solutions at the isolated singular point.At the same time,the singular solutions may exist only if the diffusion dominates the convection.

ON THE PROBLEM OF THE NUMBER OF BIFURCATION SOLUTIONS AT SINGULAR POINT

In this paper, it is proved that the solutions of a nonlinear equation are isolated under the condition that the singular points are isolated. It shows that there must have and only, have finite solutions branching from bifurcation point. This is important for the numerical analysis of bifurcation problems.

On the extension of L^2 holomorphic functions VII： Hypersurfaces with isolated singularities

An extension theorem for holomorphic functions with L^2 growth condition is strengthened for the case of the extension from hypersurfaces with isolated singularities.

Necessary and sufficient condition of C^0 flows on closed surfaces with isolated singular points having the pseudo-orbit tracing property

IN this letter we discuss the necessary and sufficient condition of C^0 flows on closed surfaces with isolated singular points having the pseudo-orbit tracing property. According to ref. [1], on a closed surface, every minimal set of a C^r(r≥2) flow is trivial, but it is possible for a C^0 flow to contain non-trivial minimal sets. Thus C^0 flows on closed surfaces are more complicated than C^r(r≥2) flows.

Thom-Sebastiani properties of Kohn-Rossi cohomology of compact connected strongly pseudoconvex CR manifolds

Let X1 and X2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann （CR） manifolds of dimensions 2m - 1 and 2n - 1 in Cm＋l and Cn＋l, respectively. We introduce the Thom- Sebastiani sum X = X1 X2 which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m＋2n＋ 1 in Cm＋n＋2. Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in C^n＋1 for all n ≥ 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X1 X2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly, we show that if X = X1 X2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X1 and X2 provided that X2 admits a transversal holomorphic Sl-action.

A NOTE ON THE STABILITY OF A CLASS OF DEGENERATE PLANAR CRITICAL POINTS

A computable expression of the asymptotic expansion of the return map for a degenerate singular point of a class of planar differential system is given, and hence the stability and the type of the singular point can be decided. These results generalize the corresponding results in [Nonlinearity, 13 （2000）, p.709].