Multiplicity Results For Two Kinds Of Kirchhnoff Type Equations

We consider the multiplicity of the Kirchhoff type elliptic problem with critical growth:where a,b,?>0,1<q<2,?(?)R4 is abounded domain with smooth boundary(?)?.By using the concentration compactness argument for PS sequences and variational methods,we obtain new existence results.Next,we also study the following fourth-order elliptic equations of Kirchhoff type:where a,b>0,we have the potential V(x):R3?R.Under certain assumptions on V(x)andf(x,u),we show the multiplicity of negative energy solutions for the above problem based on the genus properties in critical point theory.