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

.