AI 中文总结
该研究针对闭黎曼流形上的Kirchhoff-Boussinesq型方程，在对称性假设下证明了临界非线性项的多等变解与次临界非线性项的基态解存在性，还得到高阶Sobolev空间的Gagliardo-Nirenberg插值不等式与等价范数。
AI 中文摘要
我们研究闭黎曼流形$(M,g)$上形如$$\sum_{j=0}^m a_i(-Δ_g)^m u \pm \text{div}_g(\vert\nabla u\vert_g^{p-2}\nabla u) = f(u),\qquad \text{ 在 } M \text{ 上}$$的Kirchhoff-Boussinesq型方程，其中$m<\dim M/2$，$(a_0,a_1,\ldots a_m)\in C^\infty_+(M)\times [0,\infty)^{m-1}\times(0,\infty)$，$2<p\leq\frac{2\dim M}{\dim M- 2}$，且$f:\mathbb{R}\rightarrow\mathbb{R}$是具有次临界或临界Sobolev增长的超线性连续非线性项。我们简要阐释了$m=2$情形下上述方程的研究动机，它是建模弯曲弹性板动力学时定常Kirchhoff-Boussinesq方程的推广。在若干对称性假设下，我们证明了当考虑临界Sobolev非线性项时存在多个等变解，对于次临界非线性项，我们还证明了基态解的存在性。作为副产品，我们证明了一个Gagliardo-Nirenberg插值不等式，并给出了高阶Sobolev空间$H_g^m(M)$上的若干等价范数。
英文摘要
We study Kirchhoff-Boussinesq-type equations on closed Riemannian manifolds $(M,g)$ of the form \[ \sum_{j=0}^m a_i(-Δ_g)^m u \pm \text{div}_g(\vert\nabla u\vert_g^{p-2}\nabla u) = f(u),\qquad \text{ on }\ M, \] where $m<\dim M/2$, $(a_0,a_1,\ldots a_m)\in C^\infty_+(M)\times [0,\infty)^{m-1}\times(0,\infty)$, $2<p\leq\frac{2\dim M}{\dim M- 2}$ and $f:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous nonlinearity of superlinear type with subcritical or critical Sobolev growth. We briefly motivate the above equation in the case of $m=2$, as an extension of the stationary Kirchhoff-Boussinesq equation when modeling the dynamics of curved elastic plates. Under some symmetry assumptions, we prove the existence of multiple equivariant solutions when considering critical Sobolev nonlinearities and, for subcritical ones, we also prove the existence of ground-state solutions. As a byproduct, we prove a Gagliardo-Nirenberg interpolation inequality and give several equivalent norms on the higher order Sobolev space $H_g^m(M)$.
