Semi-classical limit for radial nonlinear Schrödinger equation
The authors study the Schrödinger equation \(\varepsilon^{2}\Delta - V(x)u + | u |^{p-1}u=0\) in \(\mathbb{R}^{N}\). The potential \(V\) is radially symmetric, of class \(C^{1}\), uniformly continuous and such that \(\liminf_{r\rightarrow\infty}V(r) > 0\). It is also assumed that the effective potential \(U(r)=r^{\alpha(p-1)}V(r)\), \(\alpha={2(N-1)\over{p+3}}\) is either an increasing positive function with derivative uniformly bounded away from \(0\) for \(r\geq1\) or is such that there exist \(d>0\), \(\eta>0\) with \(U(r)(r-d)>0\) for \(r>0\), \(r\not=d\) and \(U'(r)\geq \eta\) for \(r\geq d\). One proves that if \(p\in (\min(5,{{N+2}\over{N-2}}))\) then one can find a sequence \(\varepsilon_{n}\rightarrow 0\) and corresponding radial solutions \(u_{n}\) of the equation such that \(| u_{n}(r) | \leq Cr^{-\alpha}\) for all positive integers \(n\) and \(r>0\) and for some positive constant \(C\). The zeros of \(u_{n}\) become dense in a neighborhood of the origin when \(n\rightarrow \infty\). Also the approximate envelope functions of \(u_{n}\) converge to some envelope function which characterize the asymptotic behavior of the solutions. The envelope function is the unique solution of a first order differential equation subject to some additional condition.
scientific article; zbMATH DE number 2174487