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large solutions

Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

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On large solutions for fractional Hamilton–Jacobi equations
Part of
Gonzalo Dávila, Alexander Quaas, Erwin Topp
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 154 / Issue 5 / October 2024

Published online by Cambridge University Press:

11 July 2023, pp. 1313-1335

Print publication:

October 2024
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We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated with fully nonlinear elliptic equations of order $2\,s$, with $s\in (1/2,\,1)$, and a coercive gradient term with subcritical power $0< p<2\,s$. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0< p<2\,s$, involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry–Lions for the subquadratic case $1< p<2$.