Harnack type inequality and a priori estimates for solutions of a class of semilinear elliptic equations

For dimensions 3⩽n⩽63⩽n⩽6, we derive here the Harnack type inequalitymaxBRu⋅minB2Ru⩽CRn−2 for C2C2, positive solutions u ofΔu−μu+K(x)un+2n−2=0 in ball B(0,3R)B(0,3R) in RnRn where R⩽1R⩽1. Here μ>0μ>0 and the constant C=C(n,μ,|K|,|∇K|)C=C(n,μ,|K|,|∇K|). For dimension 3, we assume that K is Hölder continuous with exponent θ   with 12<θ⩽1. While for dimensions n=4,5,6n=4,5,6, assume that K∈C1K∈C1 is bounded between two positive constants and that in a neighborhood of a critical point x0x0 of K, we havec|x−x0|θ−1⩽|∇K(x)|⩽C|x−x0|θ−1c|x−x0|θ−1⩽|∇K(x)|⩽C|x−x0|θ−1 for c  , C>0C>0 and n−22⩽θ⩽n−2.As an application, a priori estimates for solutions are obtained in star shaped domains.