势在无穷远处消失的四阶椭圆方程的变号解

Sign-Changing Solutions for Fourth-Order Elliptic Equations with Potential Vanishing at Infinity

摘要: Kirchhoff方程在物理领域具有基础性地位，其解的存在性的理论研究意义日益凸显，并受到广泛关注。其中，四阶Kirchhoff方程在近年来吸引了大批学者广泛研究。本文研究一类四阶Kirchhoff型椭圆方程解的存在性：${\Delta }^{2}u-\left(1+{{\displaystyle {\int }_{{ℝ}^3}\left|\nabla u\right|}}^{2}dx\right)\Delta u+V\left(x\right)u=K\left(x\right)f\left(u\right),x\in {ℝ}^3$ ，其中$V\left(x\right)$ ，$K\left(x\right)$ 为非负连续函数，$V\left(x\right)$ 在无穷远处消失，$f$ 为准临界增长的连续函数。利用极小化方法和形变引理，本文证明了上述问题在一定的条件下存在一个变号解。文献中的最新结果得到了极大的改进和推广。

Abstract: The Kirchhoff equation holds a fundamental position in the field of physics, and the significance of studying the existence of its solutions is increasingly prominent. Its application has received widespread attention. The fourth-order elliptic equation of Kirchhoff-type has attracted the particular attention of researchers in recent years. In this paper, we investigate the existence of solutions for the following fourth-order elliptic equation of Kirchhoff-type: ${\Delta }^{2}u-\left(1+{{\displaystyle {\int }_{{ℝ}^3}\left|\nabla u\right|}}^{2}dx\right)\Delta u+V\left(x\right)u=K\left(x\right)f\left(u\right),x\in {ℝ}^3$ , where $V\left(x\right)$ , $K\left(x\right)$ are nonnegative continuous functions, $V\left(x\right)$ vanishes at infinity and $f$ is a continuous function with quasicritical growth. Using a minimization argument and the quantitative deformation lemma, we prove the problem above has a sign-changing solution. Recent results from the literature are greatly improved and extended.