Positive solutions for the Kirchhoff-type problem involving general critical growth - Part I: Existence theorem involving general critical growth

In this paper, we consider the following Kirchhoff-type problem{(a+λ∫R3|∇u|2dx+λb∫R3|u|2dx)(−Δu+bu)=f(u),inR3,u∈H1(R3),u>0,inR3, where λ≥0 is a parameter, a, b are positive constants and f reaches the critical growth. Without the Ambrosetti-Rabinowitz condition, we prove the existence of positive solutions for the Kirchhoff-type problem with a general critical nonlinearity. We also study the asymptotics of solutions as λ→0. Numerical solutions for related problems will be discussed in the second part.