一类带临界指数项的Kirchhoff型方程正基态解的存在性

Existence of Positive Ground State Solutions for a Class ofKirchhoff-type Equations with Critical Exponent

本文研究了一类带临界指数的Kirchhoff型方程,克服了临界指数项产生的困难。首先,证明方程对应的能量泛函满足局部(PS)C条件,从而获得泛函的紧性条件,再利用山路引理获得了该问题的一个正山路解;其次,应用Nehari流形的方法证明了该方程至少存在一个正基态解。该结果完善了带临界指数的Kirchhoff型方程的正基态解结果,并给出这类问题的新的可解性条件。

In this paper,a class of Kirchhoff-type equations with critical exponent is discussed,and the difficulty brought by critical exponent is solved.Firstly,the compactness condition of function is obtained by proving that the corresponding energy function satisfies the local(PS)C condition.Then,a positive mountain-pass solution is obtained by Mountain Pass Theorem.Secondly,the existence of at least one positive ground state solution for this equation is proved by taking advantage of Nehari manifold method.The results have improved the positive ground state solutions for Kirchhoff-type equation with critical exponent.Moreover,a new solvable condition of this equation is given.