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- W2912268964 abstract "We prove existence, and asymptotic behavior as r→∞, of a family of singular solutions of(1)y″+2ry′+y|y|p−1−y=0,0<r<∞,2<p≤3, which satisfy limr→0+y(r)=∞ and limr→∞y(r)=0. We also prove that the “limiting solution” of this family is the ground state. Solutions of Eq. (1) are radially symmetric standing waves of the nonlinear Schrodinger equation. A central component of our investigation is the associated integral equation(2)yk(r)=ke−rr−∫r∞trsinh(t−r)yk(t)|yk(t)|p−1dt,0<r<∞. Let 2<p≤3. For each k>0 there is a unique solution yk(r) of (2), it satisfies (1), and yk(r)∼ke−rr as r→∞. We make use of (2) to prove (I) the existence of a bounded interval (0,k⁎) such that for each k∈(0,k⁎), yk′(r)<0∀r>0, y(r)→∞ as r→0+, and yk(r)∼ke−rr as r→∞, (II) when k=k⁎, yk⁎(r) is a ground state solution satisfying 0<yk⁎(0)<∞, yk⁎′(0)=0, yk⁎′(r)<0 ∀r>0, and yk(r)∼ke−rr as r→∞. We also state related open problems for future research." @default.
- W2912268964 created "2019-02-21" @default.
- W2912268964 creator A5054020432 @default.
- W2912268964 date "2019-07-01" @default.
- W2912268964 modified "2023-10-16" @default.
- W2912268964 title "New singular standing wave solutions of the nonlinear Schrodinger equation" @default.
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- W2912268964 doi "https://doi.org/10.1016/j.jde.2019.01.031" @default.
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