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Is there any information on the Morse Index of the least energy solution of $(-\Delta)^s u= f(u) \text{ in } \Omega; u=0 \text{ in } \mathbb{R}^N-\Omega.$

I searched in google but could not found any.

The Morse index for a solution obtained by the mountain pass theorem is less than equal to one in the case of laplacian. Is it true for the fractional Laplace case. For example if we consider $f(u)\sim u^{p}$.

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  • $\begingroup$ What kind of information are you looking for? The solution $u$ may behave in an unexpected way, for example, it may have local minima in $\Omega$ even if $f > 0$. $\endgroup$ – Mateusz Kwaśnicki Aug 11 '17 at 11:44
  • $\begingroup$ The book "Nonlocal Diffusion and Applications" by C. Bucur and E. Valdinoci (through Springer) has a nice presentation on minimizers for fractional Allen-Cahn equations (Theorem 4.1.2, here your f(u)=-W'(u) where W is a double-well potential). It is delivered in the context of a broader fractional version of the De Giorgi Conjecture. I am not an expert on the connections of De Giorgi's conjecture to Morse theory, by a quick search on Google brings up an article by M. Del Pino. $\endgroup$ – SetHead Aug 11 '17 at 14:01

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