Ground states of a logarithmic Schrödinger equation with a power perturbation in $\mathbb{R}^N$: existence, uniqueness and nondegeneracy

Abstract

The objective of this paper is to study the existence and uniqueness of positive ground state solutions of the following Schrödinger problem \[ -Δu=u\log u^2+μ\abs{u}^{p-1}u\qquad\text{in }\mathbb {R}^N, \] where $N\ge2$, $1 0.$ We prove that there exists a positive ground state solution for every $μ>0$ and every such solution is smooth, radially symmetric with respect to some point, strictly decreasing in the radial variable. Moreover, as $μ\downarrow0,$ the positive ground state solutions centered at the origin will converge, uniformly over the ground-state set, to $U_0(x)=\exp(\frac{(N-\abs{x}^2)}{2})$ in the logarithmic Orlicz--Sobolev space and in $C^{2,α}(\R^N)$ for every $α\in(0,1),$ respectively. Furthermore, for sufficiently small $μ,$ we also prove that the positive ground state solutions are unique up to translations and the kernel space of the real scalar linearized operator can be spanned by the translation derivatives. The proof combines minimization on the Nehari manifold with uniform compactness and an exterior coefficient estimate that prevents normalized difference and kernel sequences from vanishing locally.