ON UNBOUNDED SOLUTIONS OF ERGODIC PROBLEMS FOR NON-LOCAL HAMILTON-JACOBI EQUATIONS

Abstract : We study an ergodic problem associated to a non-local Hamilton-Jacobi equation defined on the whole space λ − L[u](x) + |Du(x)| m = f (x) and determine whether (unbounded) solutions exist or not. We prove that there is a threshold growth of the function f , that separates existence and non-existence of solutions, a phenomenum that does not appear in the local version of the problem. Moreover, we show that there exists a critical ergodic constant, λ * , such that the ergodic problem has solutions for λ λ * and such that the only solution bounded from below, which is unique up to an additive constant, is the one associated to λ * .
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Cristina Brândle, Emmanuel Chasseigne. ON UNBOUNDED SOLUTIONS OF ERGODIC PROBLEMS FOR NON-LOCAL HAMILTON-JACOBI EQUATIONS. Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2019, 180, pp.94-128. ⟨hal-01864454⟩

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