关于RN 上非齐次四阶椭圆方程的注记

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<+∞.

ANoteonaNonhomogeneousFourth-OrderEllipticEquationonRN

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<+∞.