AI 中文总结
研究有界凸区域中分数阶半线性狄利克雷问题正弱解的一致\(L^\infty\)先验界，针对轻微次临界非线性项\(f(t) = t^q L(t)\)，通过特定结构条件防止质量集中，建立全局一致界，扩展了此类估计成立的非线性项类别。
AI 中文摘要
我们研究在有界、凸、\(C^{1,1}\)区域\(\Omega \subset \mathbb{R}^N\)中，具有齐次外部条件\(u\equiv 0\)于\(\mathbb{R}^N\setminus\Omega\)的分数阶半线性狄利克雷问题\((-\Delta)^s u = f(u)\)正弱解的一致\(L^\infty(\Omega)\)先验有界性。考虑形如\(f(t) = t^q L(t)\)的轻微超线性非线性项，其中\(1 \le q \le \frac{N + 2s}{N - 2s}\)且\(L\)是慢变函数。在严格次临界情形\(q < \frac{N + 2s}{N - 2s}\)下一致估计已确立，但轻微次临界情形\(q = \frac{N + 2s}{N - 2s}\)因可能形成泡状轮廓极具挑战。本文通过一个结构条件\(\lim_{t \to \infty} \frac{t \, |L'(t)|}{L^{\frac{N}{2s}}(t)} = \infty\)防止质量集中，从而建立了正解的全局一致\(L^\infty(\Omega)\)界，显著扩展了已知此类估计成立的非线性项类别。
英文摘要
We study the uniform $L^\infty(Ω)$ a priori boundedness of positive weak solutions to the fractional semilinear Dirichlet problem $(-Δ)^s u = f(u)$ in a bounded, convex, $C^{1,1}$ domain $Ω\subset \mathbb{R}^N$ with homogeneous exterior condition $u\equiv 0$ in $\mathbb{R}^N\setminusΩ$. 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 $L^\infty(Ω)$ bounds for positive solutions, significantly expanding the class of known nonlinearities for which such estimates hold.
Comments20 pages
