New singular standing wave solutions of the nonlinear Schrodinger equation

We prove existence, and asymptotic behavior as r -> infinity, of a family of singular solutions of y ''+2/ry' + y vertical bar y vertical bar(p-1) - y=0, 0<r<infinity, 2<p <= 3, (1) which satisfy lim(r -> 0) + y(r) = infinity and lim(r ->infinity) 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 y(k)(r) = ke(-r)/r - integral(infinity)(r) t-r sinh(t-r)yk(t)vertical bar yk(t)vertical bar(p-1) dt, 0 < r < infinity (2) Let 2 < p <= 3. For each k > 0 there is a unique solution y(k)(r) of (2), it satisfies (1), and y(k)(r) similar to ke(-r)/r as r -> infinity. We make use of (2) to prove (I) the existence of a bounded interval (0, k*) such that for each k epsilon (0, k*), y(k)'(r) < 0 for all(r) > 0, y(r) -> infinity as r -> 0(+), and y(k)'(r) similar to k e(-r)/r as r -> infinity, (II) when k = k*, yk* (r) is a ground state solution satisfying 0 < y(k)* (0) < infinity, y(k)*' (0) = 0, y(k)*' (r) < 0 for all(r) > 0, and y(k)(r) similar to k e(-r)/r as r -> infinity. We also state related open problems for future research. (C) 2019 Elsevier Inc. All rights reserved.