Scalar field equation with non-local diffusion


DOI: 10.1007/s00030-015-0328-z

Cite this article as:
Felmer, P. & Vergara, I. Nonlinear Differ. Equ. Appl. (2015) 22: 1411. doi:10.1007/s00030-015-0328-z


In this paper we are interested on the existence of ground state solutions for fractional field equations of the form
$$\left\{\begin{array}{ll} (I - \Delta)^{\alpha}u = f(x, u) & \quad {\rm in} \, I\!R^N,\\ u > 0 & \quad {\rm in} \, I\!R^N, \quad \displaystyle\lim_{|x| \to \infty}u(x) = 0,\end{array}\right.$$
where \({\alpha \in (0,1)}\) and f is an appropriate super-linear sub-critical nonlinearity. We prove regularity, exponential decay and symmetry properties for these solutions. We also prove the existence of infinitely many bound states and, through a non-local Pohozaev identity, we prove nonexistence results in the supercritical case.

Mathematics Subject Classification

35J60 35Q55 35S05 

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Departamento de Ingeniería Matemática, Centro de Modelamiento, Matemático UMR2071 CNRS-UChileUniversidad de ChileSantiagoChile

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