Multibump solutions and critical groups

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