THE NON-LINEAR SCHRODINGER EQUATION WITH NON-PERIODIC POTENTIAL: INFINITE-BUMP SOLUTIONS AND NON-DEGENERACY

The equation -epsilon(2)Delta u + F(V(x), u) = 0 is considered in R-n. It is assumed that V possesses a set of critical points B for which the values of V and (DV)-V-2 satisfy certain compactness and uniformity properties. Under appropriate conditions on F the problem is shown to possess for each b is an element of B and small epsilon > 0 a solution that concentrates at b and has detailed uniformity and decay properties. This enables the construction of solutions that concentrate at arbitrary subsets of B as epsilon -> 0. Examples are given in which B is infinite and V non-periodic.