Uniqueness and nondegeneracy of sign-changing radial solutions to an almost critical elliptic problem. (English) Zbl 1375.35025

Ao, Weiwei; Wei, Juncheng; Yao, Wei

Adv. Differ. Equ. 21, No. 11-12, 1049-1084 (2016).

Summary: We study uniqueness of sign-changing radial solutions for the following semi-linear elliptic equation \[ \Delta u-u+|u|^{p-1}u=0\quad\text{in}\;\mathbb R^N,\quad u\in H^1(\mathbb R^N), \] where \(1<p<\frac{N+2}{N-2}\), \(N\geq3\). It is well-known that this equation has a unique positive radial solution. The existence of sign-changing radial solutions with exactly \(k\) nodes is also known. However, the uniqueness of such solutions is open. In this paper, we show that such sign-changing radial solution is unique when \(p\) is close to \(\frac{N+2}{N-2}\). Moreover, those solutions are non-degenerate, i.e., the kernel of the linearized operator is exactly \(N\)-dimensional.

MSC:

35B25	Singular perturbations in context of PDEs
35J61	Semilinear elliptic equations
35Q55	NLS equations (nonlinear Schrödinger equations)