关于R^N上非齐次四阶椭圆方程的注记

A Note on a Nonhomogeneous Fourth-Order Elliptic Equation on R^N

运用临界点理论中的Ekeland变分原理研究了非齐次四阶椭圆方程Δ2u-Δu+V(x)u=f(u)+h(x)u∈H2(RN)解的存在性,其中V∈C(RN,R)满足infx∈RNV(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,并且当z<0时f(z)≡0;limz→0f(z)/z=0,limz→+∞f(z)/z=l<+∞.

By using the Ekeland’s variational principle in the critical point theory,we prove the existence of the non-trivial solutions for the nonhomogeneous Kirchhoff equationΔ2 u-Δu＋V（x）u=f（u）＋h（x） u∈H2（RN）wherein V and f meet the following conditions： V∈C（RN,R）,satisfies infx∈RN V（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;limz→0 f（z）/z=0;limz→＋∞ f（z）/z=l〈＋∞.