SEMICLASSICAL ASYMPTOTICS OF INFINITELY MANY SOLUTIONS FOR THE INFINITE CASE OF A NONLINEAR SCHRODINGER EQUATION WITH CRITICAL FREQUENCY

摘要

We consider a nonlinear Schriidinger equation with critical frequency, (P-epsilon) : epsilon(2)Delta v(x) - V(x)v(x) + vertical bar v(x)vertical bar(p-1) v(x) = 0, x is an element of R-N, and v(x) -> 0 as vertical bar v(x)vertical bar -> +infinity, for the infinite case as described by Byeon and Wang. Critical means that 0 x(0). For the semiclassical limit, epsilon -> 0, the infinite case has a characteristic limit problem, (P-inf) : Delta u(x)-P(x) u(x) + vertical bar u(x)vertical bar(p-1) u(x) = 0, x is an element of Omega, with u(x) = 0 as x is an element of Omega, where Omega subset of R-N is a smooth bounded strictly star-shaped region related to the potential V. We prove the existence of an infinite number of solutions for both the original and the limit problem via a Ljusternik-Schnirelman scheme for even functionals. Fixed a topological level k we show that v(k,epsilon), a solution of (P-epsilon), subconverges, up to a scaling, to a corresponding solution of (P-inf), and that v(k,epsilon) exponentially decays out of Omega. Finally, uniform estimates on delta Omega for scaled solutions of (P-epsilon) are obtained.