Multibump solutions and critical groups

Abstract:

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 $g(x,u)=|u|^{2^*-2}u$, $N\ge 4$ 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.