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We prove a Pohozaev-type identity for both the probleminand its harmonic extension towhen 0&lt; s &lt;1. So, our setting includes the pseudo-relativistic operatorand the results showed here are original, to the best of our knowledge. The identity is first obtained in the extension setting and then ‘translated’ into the original problem. In order to do that, we develop a specific Fourier transform theory for the fractionary operator, which lead us to define a weak solution  u  to the original problem if the identity(S)is satisfied by all. The obtained Pohozaev-type identity is then applied to prove both a result of non-existence of solution to the caseifand a result of existence of a ground state, if  f  is modeled by, for a constant  κ . In this last case, we apply the Nehari–Pohozaev manifold introduced by D. Ruiz. Finally, we inform that positive solutions ofare radially symmetric and decreasing with respect to the origin, if  f  is modeled by functions like,or.

Pohozaev-type identities for a pseudo-relativistic Schrödinger operator and applications