In engineering practice, the fourth-order elliptic equation with the biharmonic op-erator?2u+c?u=f (x, u), x∈?, 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|?? =?u|?? =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∞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, respec-tively. The main results and the proofs are different from those presented in current literature.%在工程实际中,含有双调和算子的四阶椭圆问题?2 u+c?u=f(x,u),x∈?,可用来描述悬索桥的非线性振动.当悬索桥处于平衡位置且不受外力的理想情形下,相应的边界条件为u|??=?u|??=0.本文研究了一类四阶椭圆边值问题,其中非线性项f在0处渐近线性、在∞处超二次.证明方法为下降流不变集方法,主要结果是证明了这类四阶椭圆边值问题存在一个变号解以及存在无穷多个变号解的两个定理.所得结果及其证明方法均不同于现有文献中的结果.
展开▼
