Fundamental solutions and Liouville type theorems for nonlinear integral operators. (English) Zbl 1209.45009
The authors study basic properties for a class of nonlinear integral operators related to their fundamental solutions to establish Liouville type theorems: non-existence theorems for positive entire solutions for \(\mathcal Iu \leqslant 0\) and for \(\mathcal Iu + u^p \leqslant 0, p>1\).
They prove the existence of fundamental solutions and use them, via a comparison principle, to prove the theorems for entire solutions. The non-local nature of the operators poses various difficulties in the use of the comparison techniques, since the usual values of the functions at the boundary of the domain are replaced here by the values in the complement of the domain. In particular, they are not able to prove the Hadamard Three Spheres Theorem, but they still obtain some of its consequences that are sufficient for the arguments.
They prove the existence of fundamental solutions and use them, via a comparison principle, to prove the theorems for entire solutions. The non-local nature of the operators poses various difficulties in the use of the comparison techniques, since the usual values of the functions at the boundary of the domain are replaced here by the values in the complement of the domain. In particular, they are not able to prove the Hadamard Three Spheres Theorem, but they still obtain some of its consequences that are sufficient for the arguments.
Reviewer: Kun Soo Chang (Seoul)
MSC:
| 45P05 | Integral operators |
| 45G10 | Other nonlinear integral equations |
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
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