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Some nonexistence theorems for semilinear fourth-order equations.
- Source :
- Proceedings of the Royal Society of Edinburgh: Section A: Mathematics; Jun2019, Vol. 149 Issue 3, p761-779, 19p
- Publication Year :
- 2019
-
Abstract
- In this paper, we analyse the semilinear fourth-order problem (− Δ)<superscript>2</superscript> u = g (u) in exterior domains of ℝ<superscript> N </superscript>. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δ u > 0 exist if and only if N ≥ 5 and $$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$ for some δ > 0. When only radially symmetric solutions are taken into account, we also show that the monotonicity of g is not needed in this result. Finally, we consider the same problem posed in ℝ<superscript> N </superscript> and show that if g is additionally convex and lies above a power greater than one at infinity, then all positive supersolutions u automatically verify − Δ u > 0 in ℝ<superscript> N </superscript>, and they do not exist when the previous condition fails. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 03082105
- Volume :
- 149
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Proceedings of the Royal Society of Edinburgh: Section A: Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 136844541
- Full Text :
- https://doi.org/10.1017/prm.2018.47