关于R^N上非齐次Kirchhoff方程的注记

A Note on a Nonhomogeneous Kirchhoff Equation on R^N

运用临界点理论中的Ekeland变分原理研究了非齐次Kirchhoff方程-(1+b∫RN|▽u|2dx)Δu+V(x)u=f(u)+h(x)x∈RN解的存在性,其中V∈C(RN,R)满足infNV(x)≥a1>0,这里a1>0是一个常数,更进一步,对每个M>0,meas({x∈RN:V(x)≤M})<∞,这里meas表示RN中的Lebesgue测度;f∈C(R,R+),f(0)=0,并且f(z)≡0当z<0;limz→0f(z)/z=0,limz→+∞f(z)/z=l<+∞.

The existence of the nontrivial solutions for the nonhomogeneous Kirchhoff equation-（1＋b∫RN｜▽u｜2dx）Δu＋V（x）u=f（u）＋h（x） x∈RNis proved by using the Ekeland’s variational principle in critical point theory,wherein V and f meet the following conditions： V∈C（RN,R） satisfies infx∈RNV（x）≥a1〉0,where a1 is a constant.Moreover,for every M〉0,meas（{x∈RN：V（x）≤M}）〈∞,where meas denotes the Lebesgue measure in RN;f∈C（R,R＋）,f（0）=0 and f（z）≡0 when z〈0;limz0 f（z）/z=0,limz＋∞f（z）/z=l〈＋∞.