Biharmonic equations with totally characteristic degeneracy. (English) Zbl 1459.35125
Summary: 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.
MSC:
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 58J05 | Elliptic equations on manifolds, general theory |
| 31B30 | Biharmonic and polyharmonic equations and functions in higher dimensions |
| 31B25 | Boundary behavior of harmonic functions in higher dimensions |
| 35J70 | Degenerate elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
Keywords:
biharmonic operator; semilinear degenerate elliptic equation; critical exponent; existence of solutionsReferences:
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