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

Abstract:

We consider a nonlinear Schrödinger equation Pε: ε2Δv-V(x)v+|v|p-1v=0, x∈RN, with v(x)→0, as |x|→+∞, p>1 and ε>0. We consider the finite case and critical frequency as described by Byeon and Wang, i.e., the continuous non-negative potential V verifies Z={V=0}={x0}, and, as one gets close to Z, it decays like a homogeneous positive function P. As ε↓0, the semiclassical limit problem is Pfin: Δu-P(x)u+|u|p-1u=0, x∈RN, with u(x)→0, as |x|→+∞. By a Ljusternik-Schnirelman scheme we get an infinite number of solutions for (Pε) and (Pfin), vk,ε and wk, respectively. Fixed k we prove, up to a scaling, that (a) vk,ε subconverges to wk, pointwise and in a Sobolev-like norm, (b) the energy of vk,ε converges to that of wk, and (c) a concentration property: vk,ε exponentially decays out of Z, as ε↓0.