The Existence Of Solutions For Fractional Gross-Pitaevskii With L~2-critical In R~N

This paper is concerned with the constraint minimizers of an L2-critical minimization problem of fractional Gross-Piteavskii energy functionals under an L2-critical perturbation in RN(N?1).Specif-ically,the L2-critical constraint minimization problem e(?)in RN is:(?) here the GP ennergy functional E(u)has the following form:#12 where(-?)s is the fractional laplacian,0<q<4s/N,0<S<1.b?R?b?0.the potential V(x)satisfies(V).(V).V(x)E Cloc?(RN),where ? ?(0,1),lim|x|??V(x)=? and infx?RN V(x)=0.In the first chapter,firstly we describe the research background,research object and research status of fractional Gross-Piteavskii equation(FGPE),then we give the fractional Gagliardo-Nirenberg-Sobolev(GNS)inequality and Lions lemma,finally we describes the main results.In the second chapter,firstly we prove the compactness lemma,and then we analyze that the existence and nonexistece of minimizers for energy e(?).we get the result that the problem(1.3)exists constraint minimizers if and only if 0 ??<?*:=?Q?24/N for b>0 and 0<???*for b<0;furthermore on the case where b>0,we prove that the energy e(?)?? as ???*,where b is the coefficient of the perturbation term,and above-mentioned Q(x)E HS(RN)is the unique radically symmetric positive solution in RN of the following nonlinear problem:(—?)su(x)+u(x)-|U|4s/Nu(x)=0.In the third chapter,we present a rough concertration behavior of nonnegative minimizers of e(?)as???*for b>0 and give the refined energy estimates of e(?)as???*.In the fourth chapter,we mainly carry on the summary and the prospect to the topic,which include on the basis of the properties of the potential V(x),whether constraint minimizer has a unique maximum point which converges to a global minimum of V(x),and the reason which determines the concentration rates.