The compactness theorem for inhomogeneous semilinear elliptic equations with supercritical exponents

Abstract

In this paper, we consider the following inhomogeneous semilinear equation: Δ u + k ( x ) u α + f ( x ) = 0 in Ω ⊂ R n ( n ≥ 3 ) where α ≥ n + 2 n − 2 , k ( x ) , f ( x ) ∈ C 1 ( Ω ) ; a 1 < k ( x ) < b 1 , 0 < a 1 < b 1 and | ∇ log k ( x ) | ≤ C for x ∈ Ω ¯ ; f ( x ) > 0 , ‖ f ‖ L ∞ ( Ω ) ≤ a 2 and ‖ ∇ f ‖ L ∞ ( Ω ) ≤ b 2 for some constants a 2 , b 2 > 0 . Using the monotonicity inequality and ε -regularity result, we obtain the measure estimate of the blow up set for positive smooth solutions { u i } of the above equation with { ‖ u i ‖ H 1 ( Ω ) + ‖ u i ‖ L α + 1 ( Ω ) } bounded.