The paper deals with the following Kirchhoff equations: $\begin{array}{rl}& -\left(a+b{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\mathrm{△}u+V\left(x\right)u=K\left(x\right)f\left(u\right)\mathrm{i}\mathrm{n}{\mathbb{R}}^N,\\ & u\in H^1\left({\mathbb{R}}^N\right),\end{array}$ where $N\ge 3$, $f\left(u\right)$ is $C^1$ real function satisfying quasicritical growth at infinity, and $V\left(x\right),K\left(x\right)$ are positive and continuous functions. Combining Mountain Pass Theorem and compact embeddings in weighted Sobolev spaces, we establish the existence of at least a positive and a negative solution. Moreover, using a quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution $u_0$ with two nodal domains. Finally, we show that the energy of $u_0$ is strictly larger than the ground state energy.

该论文涉及以下基尔霍夫方程： $\begin{array}{rl}& -\left(\mathrm{一种}+乙{\int }_{{\mathbb{电阻}}^N}|\mathrm{\nabla }你{|}^2\mathrm{d}X\right)\mathrm{△}你+伏\left(X\right)你=钾\left(X\right)F\left(你\right)\mathrm{一世}\mathrm{n}{\mathbb{电阻}}^N,\\ & 你\in H^1\left({\mathbb{电阻}}^N\right),\end{array}$ 在哪里 $N\ge 3$, $F\left(你\right)$ 是 $C^1$ 满足无穷远准临界增长的实函数，和 $伏\left(X\right),钾\left(X\right)$是正函数和连续函数。将 Mountain Pass 定理和加权 Sobolev 空间中的紧凑嵌入相结合，我们建立了至少一个正解和一个负解的存在。此外，使用定量变形引理，我们证明该问题具有一个最小能量符号变化的解决方案${你}_0$有两个节点域。最后，我们证明了能量${你}_0$ 严格大于基态能量。