Liouville-type theorems for polyharmonic systems in RN

By Jiaqun Liu, Yuxia Guo, Yajing Zhang. 2006

65 citations recorded by the source corpus.

In this note, we consider the polyharmonic system (−Δ)mu=vα, (−Δ)mv=uβ in RN with N>2m and α⩾1, β⩾1, where (−Δ)m is the polyharmonic operator. For 1/(α+1)+1/(β+1)>(N−2m)/N, we prove the non-existence of non-negative, radial, smooth solutions. For 1 1, we show the non-existence of non-negative smooth solutions for polyharmonic system of inequalities (−Δ)mu⩾vα, (−Δ)mv⩾uβ. More general, we can prove that all the above results hold for the system (−Δ)mu=vα,(−Δ)nv=uβ in RN with N>max{2m,2n} and α⩾1, β⩾1.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/53af23dd56cc655cc9a76b58a77f053608d35118

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In Applied Mathematics

1. L. D’Ambrosio
2. Enzo Mitidieri
3. G. Caristi
4. Serena Dipierro
5. Dušan D. Repovš
6. V. Ruadulescu
7. Tsung‐fang Wu
8. Roberta Filippucci

In Computational Theory and Mathematics

1. Enzo Mitidieri
2. L. D’Ambrosio
3. Tsung‐fang Wu
4. V. Ruadulescu
5. Dušan D. Repovš
6. Roberta Filippucci

Citations in the corpus

• Sharp criteria of Liouville type for some nonlinear systems (2013, unknown): “(1) When k ∈ (2, n/2) is an integer, the existence of the 2k-order PDEs (1.2) in the supercritical cases was obtained by Liu, Guo and Zhang [38].”
• Representation Formulae for Solutions to Some Classes of Higher Order Systems and Related Liouville Theorems (2008, background): “The following, contains as a special case Theorem 1.1 of [ 17 ], (see also [30] for related results).”