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

Abstract

For dimensions 3 ⩽ n ⩽ 6 , we derive here the Harnack type inequality max B R u ⋅ min B 2 R u ⩽ C R n − 2 for C 2 , positive solutions u of Δ u − μ u + K ( x ) u n + 2 n − 2 = 0 in ball B ( 0 , 3 R ) in R n 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 1 2 < θ ⩽ 1 . While for dimensions n = 4 , 5 , 6 , assume that K ∈ C 1 is bounded between two positive constants and that in a neighborhood of a critical point x 0 of K, we have c | x − x 0 | θ − 1 ⩽ | ∇ K ( x ) | ⩽ C | x − x 0 | θ − 1 for c, C > 0 and n − 2 2 ⩽ θ ⩽ n − 2 . As an application, a priori estimates for solutions are obtained in star shaped domains.