渐近线性基尔霍夫型方程的解的存在性

Existence of solutions for asymptotically linear Kirchhoff equations

研究基尔霍夫型方程a+λ∫R N|▽u|2+V(x)u 2[-Δu+V(x)u]=K(x)f(u),in R N,其中N≥3,a>1,λ≥0是一个参数,并且f(t)在无穷远处是渐近线性的.通过变分的方法,在对K(x)作出适当的假设下可以得到方程的非平凡解的存在性.利用截断函数得到有界的PS序列.

In this paper,we study the following Kirchhoff type equation a+λ∫R N|▽u|2+V(x)u 2[-Δu+V(x)u]=K(x)f(u),in R N,where N≥3,a>1,λ≥0 is a parameter,and f(t)is asymptotically linear at infinity.By using variational methods,we obtain the existence of nontrivial solutions under appropriate assumptions on K(x).A cut-off functional is utilized to obtain the bounded Palais-Smale sequences.