INTRODUCTION 3
In particular
(0.14) tv (λt, 1, d) = 2λvλ(t,
λ2,
d).
Remark 3. Using the contraction mapping principle it can be shown that for
every (λ, d) there exists σ(λ, d) such (0.12) has a unique positive solution defined on
the maximal interval [0, σ(λ, d)). If σ(λ, d) +∞ one has limt→σ(λ,d) v(t, λ, d) = 0
and v(t, λ, d) 0 on [0, σ(λ, d)).
For the sake of simplicity in the notation we will write
(0.15) v(t, 1, d) v(t, d) and σ(1, d) = σ(d).
The steady state solutions to (0.9) subject to (0.7) or to (0.8) are the solutions
to (0.12) satisfying
(0.16) v(1, λ, d) = 1,
or
(0.17) v (1, λ, d) = 0,
respectively. When p q, it follows by a phase plane analysis that v 1 is the only
solution of (0.12) satisfying v(1, λ, d) = 1 or v (1, λ, d) = 0 respectively. Therefore,
we will assume in the rest of this paper that p q.
Throughout this paper
(0.18) 0 μ1
2
μ2
2
· · · ∞,
and
(0.19) 0 = ν1
2
ν2
2
· · ·
will denote the radial eigenvalues of the negative Laplacian operator with zero
Dirichlet and Neumann boundary data on the unit ball in
RN
respectively. That
is, for each k = 1, . . . there exists radial functions wk = 0, zk = 0 satisfying Δwk +
μkwk = 0 and the boundary condition (0.7), and Δzk + νkzk = 0 and the boundary
condition (0.8).
For future reference we note that the μks, νks are given by the Bessel function
J defined as the solution to
(0.20) J (t) +
N 1
t
J (t) + J(t) = 0 t 0, J(0) = 1, J (0) = 0.
In fact, the μks are the zeroes of J and the νks are the zeroes of J . These numbers
also satisfy
(0.21) νk μk νk+1
for all k = 1, 2 . . ..
Our main results are the next theorems.
Theorem 1. Let q p, and
Σ = {(λ, d) (0, ∞) × (0, ∞); v(1, λ, d) = 1}.
The set of positive solutions to ( 0.9) subject to ( 0.7) is homeomorphic to Σ and is
connected. Moreover, for each positive integer j there exist 0 dj 1 and Dj 1
and a continuous function ξj : (dj, Dj) (0, ∞) such that v(·, λ, d), d = 1, is a
solution to ( 0.9) subject to ( 0.7) with v(·, λ, d) 1 having j 1 zeroes in (0, 1) if
and only if λ =
ξj(d). In addition, ξj(1) =
μj
p−q
.
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