In this paper,we investigate the existence and multiplicity of normalized solutions for the following fractional Schrödinger equations{(-△)^(s)u+λu=|u|^(p-2)u-|u|^(q-2)u,x∈R^(N),∫_(R^(N))|u|^(2)dx=c>0,wher...In this paper,we investigate the existence and multiplicity of normalized solutions for the following fractional Schrödinger equations{(-△)^(s)u+λu=|u|^(p-2)u-|u|^(q-2)u,x∈R^(N),∫_(R^(N))|u|^(2)dx=c>0,where N≥2,s∈(0,1),2+4s/N<p<q≤2_(s)^(*)=2N/N-2s,(-△)^(s)represents the fractional Laplacian operator of order s,and the frequencyλ∈R is unknown and appears as a Lagrange multiplier.Specifically,we show that there exists a c>0 such that if c>c,then the problem(P)has at least two normalized solutions,including a normalized ground state solution and a mountain pass type solution.We mainly extend the results in[Commun Pure Appl Anal,2022,21:4113–4145],which dealt with the problem(P)for the case 2<p<q<2+4s/N.展开更多
基金supported by the NNSF of China(12471103)the Natural Science Foundation of Guangdong Province(2024A1515012370)the Guangzhou Basic and Applied Basic Research(2023A04J1316)。
文摘In this paper,we investigate the existence and multiplicity of normalized solutions for the following fractional Schrödinger equations{(-△)^(s)u+λu=|u|^(p-2)u-|u|^(q-2)u,x∈R^(N),∫_(R^(N))|u|^(2)dx=c>0,where N≥2,s∈(0,1),2+4s/N<p<q≤2_(s)^(*)=2N/N-2s,(-△)^(s)represents the fractional Laplacian operator of order s,and the frequencyλ∈R is unknown and appears as a Lagrange multiplier.Specifically,we show that there exists a c>0 such that if c>c,then the problem(P)has at least two normalized solutions,including a normalized ground state solution and a mountain pass type solution.We mainly extend the results in[Commun Pure Appl Anal,2022,21:4113–4145],which dealt with the problem(P)for the case 2<p<q<2+4s/N.
