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 $\left( \textrm{P}_\varepsilon \right)$: $\varepsilon ^2 \Delta v - V(x) \ v + |v |^{p-1} \ v = 0$, $x\in \mathbb {R}^N$, with $v(x) \rightarrow 0$, as $|x |\rightarrow +\infty$, $p>1$ and $\varepsilon >0$. We consider the finite case and critical frequency as described by Byeon and Wang, i.e., the continuous non-negative potential V verifies $\mathcal {Z} = \{V = 0 \} = \{x_0\}$, and, as one gets close to $\mathcal {Z}$, it decays like a homogeneous positive function P. As $\varepsilon \downarrow 0$, the semiclassical limit problem is $\left( \textrm{P}_{\textrm{fin}} \right)$: $\Delta u - P(x) \, u + |u |^{p-1} u=0$, $x\in \mathbb {R}^N$, with $u(x)\rightarrow 0$, as $|x |\rightarrow +\infty$. By a Ljusternik-Schnirelman scheme we get an infinite number of solutions for $(\textrm{P}_\varepsilon )$ and $(\textrm{P}_{\textrm{fin}})$, $v_{k,\varepsilon }$ and $w_{k}$, respectively. Fixed k we prove, up to a scaling, that (a) $v_{k,\varepsilon }$ subconverges to $w_{k}$, pointwise and in a Sobolev-like norm, (b) the energy of $v_{k,\varepsilon }$ converges to that of $w_{k}$, and (c) a concentration property: $v_{k,\varepsilon }$ exponentially decays out of $\mathcal {Z}$, as $\varepsilon \downarrow 0$.