The purpose of this paper is to study the existence of sign-changing solution to the following fourth-order equation: 0.1 $$ \Delta ^{2}u- \biggl(a+ b \int _{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \Delta u+V(x)u=K(x)f(u) \quad\text{in } \mathbb{R}^{N}, $$ where $5\leq N\leq 7$ , $\Delta ^{2}$ denotes the biharmonic operator, $K(x), V(x)$ are positive continuous functions which vanish at infinity, and $f(u)$ is only a continuous function. We prove that the equation has a least energy sign-changing solution by the minimization argument on the sign-changing Nehari manifold. If, additionally, f is an odd function, we obtain that equation has infinitely many nontrivial solutions.

中文翻译：

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Schrödinger-Kirchhoff型四阶方程的势变解有无穷大

本文的目的是研究以下四阶方程的符号转换解的存在：0.1 $$ \ Delta ^ {2} u- \ biggl（a + b \ int _ {\ mathbb {R} ^ { N}} \ vert \ nabla u \ vert ^ {2} \ ,, dx \ biggr）\ Delta u + V（x）u = K（x）f（u）\ quad \ text {in} \ mathbb {R} ^ {N}，$$，其中$ 5 \ leq N \ leq 7 $，$ \ Delta ^ {2} $表示双谐波算子，$ K（x），V（x）$是正连续函数，在无穷远处消失， $ f（u）$只是一个连续函数。我们通过对符号变化的Nehari流形上的极小值论证证明该方程具有最小的能量符号变化解。此外，如果f是一个奇数函数，我们将得出该方程具有无限多个非平凡解。