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

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 so