We are concerned with the stability estimates for the Gagliardo–Nirenberg–Sobolev inequality: \(\begin{aligned} A\Vert \nabla u\Vert _2^\theta \Vert u\Vert _{2p}^{1-\theta } \ge \Vert u\Vert _{p+1},\ &\text { if } p\in (1/2,1),\\ A\Vert \nabla u\Vert _2^\theta \Vert u\Vert _{p+1}^{1-\theta } \ge \Vert u\Vert _{2p},\ \ \ &\text { if } p\in (1,2^*/2), \end{aligned}\) where \(2^*\) is the critical Sobolev exponent. By demonstrating the nondegeneracy of ground state solutions for the Euler–Lagrange equations corresponding to the GNS inequality, we establish stability estimates for the GNS inequality in both the sublinear case ( \(p\in (1/2,1)\) ) and the superlinear case ( \(p\in (1,2^*/2)\) ). Specifically, in the superlinear case, our findings extend previous results obtained by Nguyen (J Funct Anal 277:2179–2208, 2019), which were originally confined to \(p\in (1, (2N+1)/(2N-3))\) , to cover the entire admissible range of \(p\in (1,2^*/2)\) . Moreover, we demonstrate that there is an equivalence between the stability estimates and the nondegeneracy results.
