4 A. CASTRO AND V. PADR
´
ON
We further describe the properties of Σ in the following theorem. Let 2∗ 1 =
N+2
N−2
be the critical exponent.
Theorem 2. Let p, q, ξj as in Theorem 1. Then:
1. d1 = d2 = · · · = 0, D1 = ∞, and D2 = D3 = · · · := D.
2. If p 1 then 0 ξ1(d) c for d (1, ∞), and limd→∞ ξ1(d) = 0 if and
only if p
2∗
1.
3. If −1 q p 2∗ 1, then D ∞. If q 1 then limd→D ξj(d) = for
j = 2, 3, . . . . If q 1 then 0 ξj(d) cj for j = 2, 3, . . . , and d (1, D).
4. If q p = 1, then limd→∞ ξ1(d) = μ1, 2 and 0 limd→∞ ξj(d) for
j = 2, 3, . . . .
5. If −1 q p 1, then limd→∞ ξ1(d) = ∞.
6. If q −1 and 1 p 2∗ 1, then D = and limd→∞ ξj(d) = 0 for
j = 2, 3, . . . .
7. If p 1 and q −1 then D = and limd→∞ ξj(d) = for j = 1, 2, . . . .
8. If p 2∗ 1, then D = and 0 ξj(d) cj for j = 2, 3, . . . , and
d (1, D).
9. If q 1 then limd→0 ξj(d) = for j = 1, 2, . . . .
10. If q 1 then 0 limd→0 ξj(d) for j = 1, 2, 3, . . . .
Similarly, we have the following two theorems corresponding to the boundary
condition (0.8).
Theorem 3. Let q p, and
Ψ = {(λ, d) (0, ∞) × (0, ∞); v (1, λ, d) = 0}.
The set of positive solutions to ( 0.9) subject to ( 0.8) is homeomorphic to Ψ and
is connected. Moreover, for each positive integer j there exist 0 dj 1, Dj 1
and a continuous function ψj : (dj, Dj) (0, ∞) such that v(·, λ, d), d = 1, is a
solution to ( 0.9) subject to ( 0.8) with v(·, λ, d) 1 having j 1 zeroes in (0, 1) if
and only if λ = ψj(d). In addition, ψj(1) =
νj
p−q
.
Theorem 4. Let p, q, ψj as in Theorem 3. Then:
1. d1 = d2 = · · · = 0, D1 = ∞, and D2 = D3 = · · · := D.
2. If p 2∗ 1, then ψ1(d) = 0, d (0, ∞).
3. If q −1, then D ∞. If q 1 then limd→D ψj (d) = for j = 2, 3, . . . .
If q 1 then 0 ψj (d) cj for j = 2, 3, . . . , and d (1, D).
4. If q p = 1, then 0 limd→∞ ψj(d) for j = 2, 3, . . . .
5. If q −1 and p 1, then D = and limd→∞ ψj(d) = 0 for j = 2, 3, . . . .
6. If q −1 and p 1 then D = and limd→∞ ψj(d) = for j = 2, . . . .
7. If p
2∗
1, then D = and 0 ψj(d) cj for j = 2, 3, . . . , and
d (1, D).
8. If q 1 then limd→0 ψj(d) = for j = 2, . . . .
9. If q 1 then 0 limd→0 ψj(d) for j = 2, 3, . . . .
In chapter 2 the reader finds graphs that visualize the contents of these results.
The proof of the above theorems rely strongly on the existence and uniqueness
of the ground states for (0.9). A ground state to (0.9) is a nonnegative solution to
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