Intermediate reduction method and infinitely many positive solutions of nonlinear Schrodinger equations with non-symmetric potentials

We consider the standing-wave problem for a nonlinear Schrodinger equation, corresponding to the semilinear elliptic problem -Delta u + V(x)u = vertical bar u vertical bar(p-1)u, u epsilon H-1 (R-2), where V(x) is a uniformly positive potential and p > 1. Assuming that V(x) = V-infinity + a/vertical bar x vertical bar(m) + O (1/vertical bar x vertical bar(m+sigma)), as vertical bar x vertical bar -> +infinity, for instance if p > 2, m > 2 and sigma > 1 we prove the existence of infinitely many positive solutions. If is radially symmetric, this result was proved in [43]. The proof without symmetries is much more difficult, and for that we develop a new intermediate Lyapunov-Schmidt reduction method, which is a compromise between the finite and infinite dimensional versions of it.