4 A. CASTRO AND V. PADR ´ 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) = μ2, 1 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
Previous Page Next Page