The compactness theorem for inhomogeneous semilinear elliptic equations with supercritical exponents

In this paper, we consider the following inhomogeneous semilinear equation: Δu+k(x)uα+f(x)=0inΩ⊂Rn(n≥3) where α≥n+2n−2,k(x),f(x)∈C1(Ω); a10, ‖f‖L∞(Ω)≤a2 and ‖∇f‖L∞(Ω)≤b2 for some constants a2,b2>0. Using the monotonicity inequality and ε-regularity result, we obtain the measure estimate of the blow up set for positive smooth solutions {ui} of the above equation with {‖ui‖H1(Ω)+‖ui‖Lα+1(Ω)} bounded.