带变号势和对数非线性项Kirchhoff问题解的存在性

The existence of the solutions for Kirchhoff problems with sign-changing potential and logarithmic nonlinearity

讨论了带变号势和对数非线性项Kirchhoff问题的非平凡解.通常Kirchhoff问题的非线性项只是多项式形式,然而本文所处理的非线性项是对数非线性项.由对数型Sobolev不等式可知带变号势和对数非线性项Kirchhoff问题的能量泛函满足山路型结构,再利用序列的有界性得到了PS条件,最后结合山路定理,获得了该问题非平凡解的存在性结论.

In this paper,we discuss the existence of nontrivial solutions to Kirchhoff problem with sign-changing potential and logarithmic nonlinearity.The nonlinearity of Kirchhoff problem is usually polynomial,but the nonlinearity in this paper is logarithmic nonlinearity.The energy functional of Kirchhoff problem with sign-changing potential and logarithmic nonlinearity satisfies the mountain geometry structure by logarithmic Sobelev inequality,the PS condition is obtained through bounded sequence and the existence of nontrivial solutions is obtained by mountain pass theorem.