A priori bounds for positive solutions of Kirchhoff type equations

Let Ω be a bounded smooth domain in RN. Assume that 0<α<2∗−12, a>0, and b>0. We consider the following Dirichlet problem of Kirchhoff type equation (0.1)−(a+b‖∇u‖22α)Δu=|u|p−1u+h(x,u,∇u)inΩ,u=0on∂Ωwith p∈(0,2∗)∖{1}. Where 2∗=+∞ for N=2, and 2∗=N+2N−2 for N≥3. Under suitable conditions of h(x,u,∇u) (see (A), (H1) and (H2) in Section 3), we get a priori estimates for positive solutions to problem (0.1). By making use of these estimates and the continuous method, we further get some existence results for positive solutions to problem (0.1) when 0