| Abstract: |
| We study nonlinear fractional equations $$(-\Delta)^s u(x) = f(u(x))$$ in a half-space and prove that all positive solutions are strictly increasing in the $x_n$-direction.
Previous results typically require the solution $u$ to be globally bounded in $\mathbb{R}^n$. We substantially weaken this condition by assuming only that $u$ be bounded in each slab.
As a crucial ingredient, we obtained a boundary H\{older} regularity estimate that requires only the boundedness of $u$ near the boundary. This represents a significant improvement over existing results, which often assumed global boundedness of $u$ throughout $\mathbb{R}^n$.
To derive the monotonicity, we employ the method of moving planes. We first obtain a narrow region principle in unbounded domains, then we apply {\em averaging effects} four times to ensure that the planes can be moved continuously all the way to $x_n = \infty$. Compared with the traditional approaches, methods based on this new technique-{\em the averaging effect} require substantially weaker regularity assumptions and can even accommodate unbounded solutions. |
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