We consider the Newtonian system −¨q + B(t)q = Wq(q, t) with B, W periodic in t, B positive definite, and show that for each isolated homoclinic solution q0 having a nontrivial critical group (in the sense of Morse theory), multibump solutions (with 2 ≤ k ≤∞ bumps) can be constructed by gluing translates of q0. Further we show that the collection of multibumps is semiconjugate to the Bernoulli shift. Next we consider the Schr¨odinger equation −Δu + V (x)u = g(x, u) in RN, where V , g are periodic in x1, . . . , xN, σ(−Δ+V ) ⊂ (0,∞), and we show that similar results hold in this case as well. In particular, if g(x, u) = |u|2∗−2u, N ≥ 4 and V changes sign, then there exists a solution minimizing the associated functional on the Nehari manifold. This solution gives rise to multibumps if it is isolated.
Multibump solutions and critical groups
ARIOLI, GIANNI;
2009-01-01
Abstract
We consider the Newtonian system −¨q + B(t)q = Wq(q, t) with B, W periodic in t, B positive definite, and show that for each isolated homoclinic solution q0 having a nontrivial critical group (in the sense of Morse theory), multibump solutions (with 2 ≤ k ≤∞ bumps) can be constructed by gluing translates of q0. Further we show that the collection of multibumps is semiconjugate to the Bernoulli shift. Next we consider the Schr¨odinger equation −Δu + V (x)u = g(x, u) in RN, where V , g are periodic in x1, . . . , xN, σ(−Δ+V ) ⊂ (0,∞), and we show that similar results hold in this case as well. In particular, if g(x, u) = |u|2∗−2u, N ≥ 4 and V changes sign, then there exists a solution minimizing the associated functional on the Nehari manifold. This solution gives rise to multibumps if it is isolated.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
