关键词: 四阶椭圆边值问题, 解的存在性, 变号解, 临界点

Abstract: In engineering practice, the fourth-order elliptic equation with the biharmonic operator $\Delta^2 u + c \Delta u = f(x,u), x \in \Omega$, can be used to describe the deformation of an suspension bridge. When the bridge is in equilibrium and there are no external forces, the corresponding equation satisfies the boundary condition $u|_{\partial \Omega} = \Delta u|_{\partial \Omega} = 0$. In this paper, a class of fourth-order elliptic boundary value problems is examined under the assumption that the nonlinear term $f$ is asymptotically linear at $0$ and superquadric at $\infty$ with respect to $u$. The proof method is the descending flow invariant set method. The main results are two theorems which establish the existence of one sign-changing solution and infinitely many sign-changing solutions, respectively. The main results and the proofs are different from those presented in current literature.

Key words: fourth-order elliptic boundary value problems, existence of solution, sign-changing solutions, critical points