RN上一类非线性Schrödinger-Kirchhoff型方程非平凡解的存在性和多重性

摘要

利用变分方法研究了 RN上一类带有次临界非线性项的 Schr?dinger-Kirchhoff 型方程非平凡解的存在性和多重性.在一定的假设条件下,首先证明了该问题的能量泛函下方有界且满足条件,从而得到了泛函的一个临界值,于是证明了该问题至少存在一个非平凡解.进一步当非线性项为奇函数时,利用亏格性质证明了该问题存在无穷多个非平凡解.%The existence and multiplicity of nontrivial solutions for Schr?dinger-Kirchhoff type equations in RNwith subcritical nonlinearities term were investigated by using variational methods.Firstly,it was proved that the energy functional is bounded from below and satisfies (PS)condition under certain assumptions,thus giving a critical value of the energy functional. Then it was proved that the equation has at least a nontrivial solution.Furtherly,it was proved that the equation has infinitely many nontrivial solutions by combining with genus properties when the nonlinear term is odd.