For dimensions 3 ≤ n ≤ 6, we derive here the Harnack type inequalityunder(max, BR) u ṡ under(min, B2 R) u ≤ frac(C, Rn - 2) for C2, positive solutions u ofΔ u - μ u + K (x) ufrac(n + 2, n - 2) = 0 in ball B (0, 3 R) in Rn where R ≤ 1. Here μ > 0 and the constant C = C (n, μ, | K |, | ∇ K |). For dimension 3, we assume that K is Hölder continuous with exponent θ with frac(1, 2) < θ ≤ 1. While for dimensions n = 4, 5, 6, assume that K ∈ C1 is bounded between two positive constants and that in a neighborhood of a critical point x0 of K, we havec | x - x0 |θ - 1 ≤ | ∇ K (x) | ≤ C | x - x0 |θ - 1 for c, C > 0 and frac(n - 2, 2) ≤ θ ≤ n - 2. As an application, a priori estimates for solutions are obtained in star shaped domains. © 2007 Elsevier Inc. All rights reserved.
Lin, C. S., & Prajapat, J. V. (2008). Harnack type inequality and a priori estimates for solutions of a class of semilinear elliptic equations. Journal of Differential Equations, 244(3), 649–695. https://doi.org/10.1016/j.jde.2007.09.012