A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrodinger equations

The classical critical Trudinger-Moser inequality in R^(2)under the constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for anyτ>0,it holds that sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))and 4πis sharp.However,if we consider the less restrictive constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1,where V(x)is nonnegative and vanishes on an open set in R^(2),it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π.The loss of a positive lower bound of the potential V(x)makes this problem become fairly nontrivial.The main purpose of this paper is twofold.We will first establish the Trudinger-Moser inequality sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))when V is nonnegative and vanishes on an open set in R^(2).As an application,we also prove the existence of ground state solutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u)in R^(2),(0.1)where V(x)≥0 and vanishes on an open set of R^(2)and f has critical exponential growth.Having a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential)has been the standard assumption when one deals with the existence of solutions to the above Schrodinger equations when the nonlinear term has the exponential growth.Our existence result seems to be the first one without this standard assumption.

Hardy–Littlewood–Sobolev Inequalities with the Fractional Poisson Kernel and Their Applications in PDEs

The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel on the upper half space ■ where f ∈ L^p(?R_+~n), g ∈ Lq(R_+~n) and p, q'∈(1, +∞), 2 ≤α < n satisfying (n-1)/np+1/q'+(2-α)/n= 1.Second, we utilize the technique combining the rearrangement inequality and Lorentz interpolation to show the attainability of best constant C_(n,α,p,q'). Third, we apply the regularity lifting method to obtain the smoothness of extremal functions of the above inequality under weaker assumptions. Furthermore,in light of the Pohozaev identity, we establish the sufficient and necessary condition for the existence of positive solutions to the integral system of the Euler–Lagrange equations associated with the extremals of the fractional Poisson kernel. Finally, by using the method of moving plane in integral forms, we prove that extremals of the Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel must be radially symmetric and decreasing about some point ξ_0 ∈ ?R_+~n. Our results proved in this paper play a crucial role in establishing the Stein–Weiss inequalities with the Poisson kernel in our subsequent paper.

Special Issue in Honor of Professor Carlos E. Kenig

It is our great honor and pleasure to be the guest editors for this special issue of the Acta Mathematica Sinica-English Series, a flagship journal of the Chinese Mathematical Society, dedicated to Professor Carlos E. Kenig from the University of Chicago on the occasion of his 65th birthday.