ABSTRACT

Consider a class of modified Kirchhoff-type equations $-(1+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\mathrm{d}x)\mathrm{\Delta }u+V\left(x\right)u-\frac{1}{2}u\mathrm{\Delta }\left(u^2\right)=f\left(u\right),\text{in}{\mathbb{R}}^3,$ where the nonlinear term f is 4-superlinear at infinity. By using the method of invariant sets of descending flow, the existence of a sign-changing solution is obtained. When f is assumed to be odd, we prove that the above problem admits infinitely many sign-changing solutions. Moreover, when $V\left(x\right)\equiv 1$ and the nonlinearity of power growth $f\left(u\right)=|u{|}^{p-2}u$ with 1<p, we establish some existence and non-existence results. For $p\in (1,2]\cup [12,\mathrm{\infty })$, the non-existence result relies on the deduction of some suitable Pohozaev identity. For $p\in (3,4]$, using the Nehari–Pohozaev manifold, we prove that the above problem admits a ground state solution.

摘要

考虑一类修正的基尔霍夫型方程 $-(1+乙{\int }_{{\mathbb{电阻}}^3}|\mathrm{\nabla }你{|}^2\mathrm{d}X)\mathrm{\Delta }你+伏\left(X\right)你-\frac{1}{2}你\mathrm{\Delta }\left({你}^2\right)=F\left(你\right),\text{在}{\mathbb{电阻}}^3,$其中非线性项f在无穷远处是 4-超线性的。利用下降流不变集的方法，得到了变号解的存在性。当f被假定为奇数时，我们证明上述问题允许无限多个变号解。此外，当$伏\left(X\right)\equiv 1$ 和功率增长的非线性 $F\left(你\right)=|你{|}^{p-2}你$当 1< p 时，我们建立了一些存在和不存在的结果。为了$p\in (1,2]\cup [12,\mathrm{\infty })$，不存在的结果依赖于一些合适的 Pohozaev 恒等式的推导。为了$p\in (3,4]$，使用 Nehari-Pohozaev 流形，我们证明上述问题允许基态解。