2026/07/21 by Juan-Carlos Felipe-Navarro, Rosa Pardo
#math.AP
We study the uniform L^∞(Ω) a priori boundedness of positive weak solutions to the fractional semilinear Dirichlet problem (-Δ)s u = f(u) in a bounded, convex, C1,1 domain Ω⊂ ℝN with homogeneous exterior condition u≡ 0 in ℝN∖Ω. We consider slightly superlinear nonlinearities of the form f(t) = tq L(t), where 1 ≤ q ≤ (N+2s)/(N-2s) and L is a slowly varying function. Although uniform estimates are well-established in the strictly subcritical regime q < (N+2s)/(N-2s), the slightly subcritical case, q = (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 limt → ∞ \fract |L'(t)|L(N)/(2s)(t) = ∞, which acts as an asymptotic barrier that prevents mass concentration. Under this assumption, we establish global uniform L^∞(Ω) bounds for positive solutions, significantly expanding the class of known nonlinearities for which such estimates hold.