We study the existence of standing waves of fractional Schrodinger equations with a potential term and a general nonlinear term:iut-(-Δ)^(s)u-V(x)u+f(u)=0,(t,x)∈R_(+)×R^(N),where s∈(0,1),N>2s is an integer ...We study the existence of standing waves of fractional Schrodinger equations with a potential term and a general nonlinear term:iut-(-Δ)^(s)u-V(x)u+f(u)=0,(t,x)∈R_(+)×R^(N),where s∈(0,1),N>2s is an integer and V(x)≤0 is radial.More precisely,we investigate the minimizing problem with L2-constraint:E(a)=inf{1/2∫_(R_(N))|(-△)^(s/2)u|^(2)+V(x)|u|^(2)-2F(|u|)|u∈H^(s)(R^(N)),||u||_(L^(2))^(2)(R^(N))=α.Under general assumptions on the nonlinearity term f(u)and the potential term V(x),we prove that there exists a constant a00 such that E(a)can be achieved for all a>a_(0),and there is no global minimizer with respect to E(a)for all 0<a<a_(0).Moreover,we propose some criteria determining a0=0 or a_(0)>0.展开更多
基金funded by Natural Science Foundation of Hebei Province(No.A2022205007)Science and Technology Project of Hebei Education Department(No.QN2022047)+1 种基金Science Foundation of Hebei Normal University(No.L2021B05)supported by National Natural Science Foundation of China(Nos.11771428,12031015 and 12026217).
文摘We study the existence of standing waves of fractional Schrodinger equations with a potential term and a general nonlinear term:iut-(-Δ)^(s)u-V(x)u+f(u)=0,(t,x)∈R_(+)×R^(N),where s∈(0,1),N>2s is an integer and V(x)≤0 is radial.More precisely,we investigate the minimizing problem with L2-constraint:E(a)=inf{1/2∫_(R_(N))|(-△)^(s/2)u|^(2)+V(x)|u|^(2)-2F(|u|)|u∈H^(s)(R^(N)),||u||_(L^(2))^(2)(R^(N))=α.Under general assumptions on the nonlinearity term f(u)and the potential term V(x),we prove that there exists a constant a00 such that E(a)can be achieved for all a>a_(0),and there is no global minimizer with respect to E(a)for all 0<a<a_(0).Moreover,we propose some criteria determining a0=0 or a_(0)>0.
