  In 2013, R.L. Frank and E. Lenzmann [R.L. Frank, E. Lenzmann, Uniqueness of
non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math.
210 (2) (2013) 261-318] study the following problem:
  \begin{align*}
  (-\Delta)^su + u = u^{p-1} \quad \text{in } \mathbb{R}^N,
  \end{align*} where $s \in (0,1)$, $N = 1$, $p \in (2,2_s^*)$, and $2_s^*$ is
the critical fractional Sobolev exponent. They proved that the ground state is
unique (up to translations). Then in 2016, they, together with L. Silvestre
[R.L. Frank, E. Lenzmann, L. Silvestre, Uniqueness of radial solutions for the
fractional Laplacian, Comm. Pure Appl. Math. 69 (9) (2016) 1671-1726] showed
similar uniqueness results for high dimensions ($N \geqslant 2$), in which they
proposed a challenging open problem to extend their results about
non-degeneracy and uniqueness of ground states to nonlinearities $f(u)$ beyond
the pure-power case. To the best of our knowledge, this question is still
unresolved so far. In this paper, we aim to give a full affirmative answer to
this open issue for a large class of convex nonlinearities. In fact, we prove
the uniqueness of the positive solution.
