REGULARIZATION OF PLANAR VORTICES FOR THE INCOMPRESSIBLE FLOW

In this paper, we continue to construct stationary classical solutions for the incompressible planar flows approximating singular stationary solutions of this problem. This procedure is carried out by constructing solutions for the following elliptic equations{-△u=λ∑1Bδ（x0,j）（u-kj）p＋,in Ω,u=0,onΩ is a bounded simply-connected smooth domain, ki （i = 1,… , k） is prescribed positive constant. The result we prove is that for any given non-degenerate critical pointX0=（x0,1,…,x0,k of the Kirchhoff-Routh function defined on Ωk corresponding to （ k1,……kk ）there exists a stationary classical solution approximating stationary k points vortex solution. Moreover, as λ→＋∞ shrinks to {x05}, and the local vorticity strength near each x0,j approaches kj, j = 1,… , k. This result makes the study of the above problem with p _〉 0 complete since the cases p 〉 1, p = 1, p = 0 have already been studied in [11, 12] and [13] respectively.

New existence of multi-spike solutions for the fractional Schrodinger equations

We consider the following fractional Schr¨odinger equation:(-Δ)^(s)u+V(y)u=u^(p);u>0 in R^(N);(0.1)where s ∈(0,1),1<p<N+2s/N-2s,and V(y)is a positive potential function and satisfies some expansion condition at infinity.Under the Lyapunov-Schmidt reduction framework,we construct two kinds of multi-spike solutions for(0.1).The first k-spike solution uk is concentrated at the vertices of the regular k-polygon in the(y1;y2)-plane with k and the radius large enough.Then we show that uk is non-degenerate in our special symmetric workspace,and glue it with an n-spike solution,whose centers lie in another circle in the(y3;y4)-plane,to construct infinitely many multi-spike solutions of new type.The nonlocal property of(-Δ)^(s)is in sharp contrast to the classical Schr¨odinger equations.A striking difference is that although the nonlinear exponent in(0.1)is Sobolev-subcritical,the algebraic(not exponential)decay at infinity of the ground states makes the estimates more subtle and difficult to control.Moreover,due to the non-locality of the fractional operator,we cannot establish the local Pohozaev identities for the solution u directly,but we address its corresponding harmonic extension at the same time.Finally,to construct new solutions we need pointwise estimates of new approximate solutions.To this end,we introduce a special weighted norm,and give the proof in quite a different way.

Achievability of a supremum for the Hardy-Littlewood-Sobolev inequality with supercritical exponent

In this paper, we prove that the supremum sup{ ∫B∫B|u（y）|p（|y|）|u（x）|p（|x|）/|x-y|μdxdy : u ∈ H0,rad1（B）, ||?||uL2（B）= 1}is attained, where B denotes the unit ball in RN（N ≥3）, μ ∈（0, N）, p（r） = 2μ*+ rt, t ∈（0, min{N/2-μ/4, N-2}） and 2μ*=（2N-μ）/（N-2） is the critical exponent for the Hardy-Littlewood-Sobolev inequality.