Vol. 104, No. 2, 1983

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Criteria for oscillatory sublinear Schrödinger equations

Charles Andrew Swanson

Vol. 104 (1983), No. 2, 483–493
Abstract

The semilinear Schrödinger equation

Lu ≡ Δu + f(x,u) = 0, x ∈ Ω
(1)

will be considered in an exterior domain Ω Rn, n 2, where f is nonnegative and locally Hölder continuous in Ω × (0,). One objective is to find sharp necessary conditions for (1) to be oscillatory in Ω under the sublinear hypothesis that max|x|=rt1f(x,t) is a nonincreasing function of t in (0,) for each fixed r > 0. The necessary conditions below are proved in §2:

r max |x|=rf(x,clog r)dr = +  if n = 2;
r max |x|=rf(x,c)dr = +  if n 3
for some positive constant c. Sufficient conditions for (1) to be oscillatory in Ω are proved in §3 under a modified sublinear hypothesis. These results are then combined to yield characterizations of oscillatory sublinear equations of the Emden-Fowler type in exterior domains.

Mathematical Subject Classification 2000
Primary: 35B05
Secondary: 35J10
Milestones
Received: 18 September 1980
Published: 1 February 1983
Authors
Charles Andrew Swanson