In this paper, we study the biharmonic operator on a manifold with conical singularities and consider the existence of non-trivial weak solution for the corresponding semilinear degenerate elliptic equation with critical cone Sobolev exponents by the cone Sobolev inequality and Poincaré inequality. Furthermore, we show an interesting integral identity, which can be used to discuss the non-existence results for some biharmonic equations under Navier boundary conditions.

中文翻译：

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具有完全特征退化的双调和方程

在本文中，我们研究了锥奇异流形上的双调和算子，并通过锥Sobolev不等式和Poincaré不等式考虑了具有临界锥Sobolev指数的半线性退化椭圆型方程的非平凡弱解的存在。此外，我们展示了一个有趣的积分恒等式，可以用来讨论Navier边界条件下某些双调和方程的不存在结果。