The compactness theorem for inhomogeneous semilinear elliptic equations with supercritical exponents Original Research Article

In this paper, we consider the following inhomogeneous semilinear equation: View the MathML sourceΔu+k(x)uα+f(x)=0inΩ⊂Rn(n≥3) Turn MathJax on where View the MathML sourceα≥n+2n−2,k(x),f(x)∈C1(Ω); a10f(x)>0, ‖f‖L∞(Ω)≤a2‖f‖L∞(Ω)≤a2 and ‖∇f‖L∞(Ω)≤b2‖∇f‖L∞(Ω)≤b2 for some constants View the MathML sourcea2,b2>0. Using the monotonicity inequality and εε-regularity result, we obtain the measure estimate of the blow up set for positive smooth solutions {ui}{ui} of the above equation with {‖ui‖H1(Ω)+‖ui‖Lα+1(Ω)}{‖ui‖H1(Ω)+‖ui‖Lα+1(Ω)} bounded.