Multibump solutions and critical groups

We consider the Newtonian system − q ¨ + B ( t ) q = W q ( q , t ) -\ddot q+B(t)q = W_q(q,t) with B B , W W periodic in t t , B B positive definite, and show that for each isolated homoclinic solution q 0 q_0 having a nontrivial critical group (in the sense of Morse theory), multibump solutions (with 2 ≤ k ≤ ∞ 2\le k\le \infty bumps) can be constructed by gluing translates of q 0 q_0 . Further we show that the collection of multibumps is semiconjugate to the Bernoulli shift. Next we consider the Schrödinger equation − Δ u + V ( x ) u = g ( x , u ) -\Delta u+V(x)u = g(x,u) in R N \mathbb {R}^N , where V V , g g are periodic in x 1 , … , x N x_1,\ldots ,x_N , σ ( − Δ + V ) ⊂ ( 0 , ∞ ) \sigma (-\Delta +V)\subset (0,\infty ) , and we show that similar results hold in this case as well. In particular, if g ( x , u ) = | u | 2 ∗ − 2 u g(x,u)=|u|^{2^*-2}u , N ≥ 4 N\ge 4 and V 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.