Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities

In this paper, we consider the following nonlinear Schrödinger equations with mixed nonlinearities:{−Δu=λu+μ|u|q−2u+|u|2⁎−2uin RN,u∈H1(RN),∫RNu2=a2, where N≥3, μ>0, λ∈R and 2<q<2⁎=2NN−2. We prove in this paper(1)Existence of solutions of mountain-pass type for N≥3 and 2<q<2+4N;(2)Exis...

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Bibliographic Details
Published inJournal of functional analysis Vol. 283; no. 6; p. 109574
Main Authors Wei, Juncheng, Wu, Yuanze
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.09.2022
Subjects
Critical nonlinearity
Ground state
Mixed nonlinearity
Normalized solution
35J20
35B33
Critical nonlinearity
Normalized solution
Ground state
35B40
Mixed nonlinearity
35B09
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Summary:In this paper, we consider the following nonlinear Schrödinger equations with mixed nonlinearities:{−Δu=λu+μ|u|q−2u+|u|2⁎−2uin RN,u∈H1(RN),∫RNu2=a2, where N≥3, μ>0, λ∈R and 2<q<2⁎=2NN−2. We prove in this paper(1)Existence of solutions of mountain-pass type for N≥3 and 2<q<2+4N;(2)Existence and nonexistence of ground states for 2+4N≤q<2⁎ with μ>0 large;(3)Precisely asymptotic behaviors of ground states and mountain-pass solutions as μ→0 and μ goes to its upper bound. Our studies answer some open questions proposed by Soave in [48].
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2022.109574