一类渐进线性三调和椭圆方程非平凡解的存在性

The Existence of Nontrivial Solution to a Class Triharmonic Elliptic Equations with Asymptotically Linear

摘要: 这篇文章考虑下列一类渐进线性三调和椭圆方程：$\{\begin{array}{ll}-{\Delta }^{3}u+c_1{\Delta }^{2}u+c_2\Delta u=f\left(x,u\right),\hfill & 在\Omega 中,\hfill \\ u=\Delta u={\Delta }^{2}u=0,\hfill & 在\partial \Omega 上,\hfill \end{array}$ ，其中${\left(-\Delta \right)}^3(\cdot )=-\Delta \left({\Delta }^2(\cdot )\right)$ 表示三调和算子，$\Omega \subset R^n\left(n\ge 1\right)$ 是一个有界的光滑区域，$c_1,c_2$ 是常数，$f\left(x,t\right)$ 对$t$ 在无穷远处是渐进线性的。通过使用山路定理，得出上述方程非平凡解的存在性。

Abstract: This article considers a class sixth-order elliptic equations with asymptotically linear: $\{\begin{array}{ll}-{\Delta }^{3}u+c_1{\Delta }^{2}u+c_2\Delta u=f\left(x,u\right),\hfill & \text{in}\Omega ,\hfill \\ u=\Delta u={\Delta }^{2}u=0,\hfill & \text{on}\partial \Omega ,\hfill \end{array}$ , where${\left(-\Delta \right)}^3(\cdot )=-\Delta \left({\Delta }^2(\cdot )\right)$ denotes the triharmonic operator, $\Omega \subset R^n\left(n\ge 1\right)$ is a smooth bounded domain,$c_1,c_2$ are constants.$f\left(x,t\right)$ asymptotically linear with respect to $t$ . By using the mountain pass theorem, we give the existence result for nontrivial solutions for the above equation.