TY - JOUR

T1 - Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities

AU - Ikoma, Norihisa

AU - Miyamoto, Yasuhito

N1 - Funding Information:
The first author was supported by JSPS KAKENHI Grant Numbers JP16K17623 and JP17H02851. The second author was supported by JSPS KAKENHI Grant Numbers 16K05225 and 17KK0086.
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/4/1

Y1 - 2020/4/1

N2 - The existence and nonexistence of the minimizer of the L2-constraint minimization problem e(α):=inf{E(u)|u∈H1(RN),∥u∥L2(RN)2=α} are studied. Here, E(u):=12∫RN|∇u|2+V(x)|u|2dx-∫RNF(|u|)dx,V(x) ∈ C(RN) , 0 ≢ V(x) ≤ 0 , V(x) → 0 (| x| → ∞) and F(s)=∫0sf(t)dt is a rather general nonlinearity. We show that there exists α≥ 0 such that e(α) is attained for α> α and e(α) is not attained for 0 < α< α. We study differences between the cases V(x) ≢ 0 and V(x) ≡ 0 , and obtain sufficient conditions for α= 0. In particular, if N= 1 , 2 , then α= 0 , and hence e(α) is attained for all α> 0.

AB - The existence and nonexistence of the minimizer of the L2-constraint minimization problem e(α):=inf{E(u)|u∈H1(RN),∥u∥L2(RN)2=α} are studied. Here, E(u):=12∫RN|∇u|2+V(x)|u|2dx-∫RNF(|u|)dx,V(x) ∈ C(RN) , 0 ≢ V(x) ≤ 0 , V(x) → 0 (| x| → ∞) and F(s)=∫0sf(t)dt is a rather general nonlinearity. We show that there exists α≥ 0 such that e(α) is attained for α> α and e(α) is not attained for 0 < α< α. We study differences between the cases V(x) ≢ 0 and V(x) ≡ 0 , and obtain sufficient conditions for α= 0. In particular, if N= 1 , 2 , then α= 0 , and hence e(α) is attained for all α> 0.

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U2 - 10.1007/s00526-020-1703-0

DO - 10.1007/s00526-020-1703-0

M3 - Article

AN - SCOPUS:85079392541

VL - 59

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 2

M1 - 48

ER -