Uniform a priori estimates for slightly subcritical fractional problems

Abstract

We study the uniform $L^{\infty }(\hat{I}\text{c◯})$ a priori boundedness of positive weak solutions to the fractional semilinear Dirichlet problem $(-\hat{I}”{)}^{s}u=f(u)$ in a bounded, convex, $C^{1,1}$ domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ with homogeneous exterior condition $u\equiv 0$ in ${\mathbb{R}}^N\setminus \hat{I}\text{c◯}$. 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 $$\underset{t\to \infty }{\lim }\frac{t|L^{\prime }(t)|}{L^{\frac{N}{2s}}(t)}=\infty ,$$ which acts as an asymptotic barrier that prevents mass concentration. Under this assumption, we establish global uniform $L^{\infty }(\hat{I}\text{c◯})$ bounds for positive solutions, significantly expanding the class of known nonlinearities for which such estimates hold.

20 pages