Uniform a priori estimates for slightly subcritical fractional problems

We study the uniform $L^\infty(\Omega)$ a priori boundedness of positive weak solutions to the fractional semilinear Dirichlet problem $(-\Delta)^s u = f(u)$ in a bounded, convex, $C^{1,1}$ domain $\Omega \subset \mathbb{R}^N$ with homogeneous exterior condition $u\equiv 0$ in $\mathbb{R}^N\setminus\Omega$. We consider slightly superlinear nonlinearities of the form $f(t) = t^q L(t)$, where $1 \le q \le \frac{N+2s}{N-2s}$ and $L$ is a slowly varying function. Although uniform estimates are well-established in the strictly subcritical regime $q < \frac{N+2s}{N-2s}$, the slightly subcritical case, $q = \frac{N+2s}{N-2s}$, is highly challenging due to the potential formation of bubbling profiles.
In this work, we isolate a structural condition on the slowly varying perturbation, namely $$ \lim_{t \to \infty} \frac{t \, |L'(t)|}{L^{\frac{N}{2s}}(t)} = \infty, $$ which acts as an asymptotic barrier that prevents mass concentration. Under this assumption, we establish global uniform $L^\infty(\Omega)$ bounds for positive solutions, significantly expanding the class of known nonlinearities for which such estimates hold.