Asymptotic Behaviour of Infinitely Many Solutions for the Finite Case of a Nonlinear Schrodinger Equation with Critical Frequency

We consider a nonlinear Schrodinger equation (P-e): e(2)?v - V(x) v + IvI(p-1) v = 0, x ?R-N, with v(x) ? 0, as Ix' ? +8, p > 1 ande > 0 . We consider the finite case and critical frequency as described by Byeon and Wang, i.e., the continuous non-negative potential Vveri-function P. As E ?0, the semiclassical limit problem is (Pfin fies Z = {V = 0} = {x(0)} , and, as one gets close to Z , it decays like a homogeneous positive ): ?u - P(x) u + IuI(p-1)u = 0, x ? R-N, with u(x) ? 0, as Ix' ? +8. By a Ljusternik-Schnirelman scheme we get an infinite number of solutions for (Pe) and (P-fin), v(k,e )and wk , respectively. Fixed k we prove, up to a scaling, that (a) v(k,e) subconverges to w(k) , pointwise and in a Sobolev-like norm, (b) the energy of v(k,e) converges to that of wk , and (c) a concentration property: v(k,e )exponentially decays out of Z, as E ?0.