What is meant by a classical solution of a fractional laplacian in $ (-\Delta)^su= f(u)\text{ in } \mathbb{R}^N$ with no condition at infinity. If one can show that u is a weak solution of the above solution, how do one show it is classical.

  • $\begingroup$ I would guess that "classical" means "pointwise"; I am not sure what would "no condition at infinity" mean. Can you provide a broader context? $\endgroup$ Mar 22 '18 at 10:16
  • $\begingroup$ I thought any bounded weak solution is a classical solution (smooth in space and time) $\endgroup$ Mar 22 '18 at 11:15
  • $\begingroup$ By "no condition at infinity", the solution may be bounded or does not decay at infinity. $\endgroup$
    – sadiaz
    Mar 23 '18 at 15:55

Usually authors define a classical solutions as a continuous function which satisfies your equation in the pointwise sense i.e for every x. Conditions at infinity often motivate the proof technique used to show existence of solutions, for instance if f is sublinear at infinity i.e is dominated by a line, sub-and supersolution method may work. If f is superlinear at infinity, i.e domninates a line, then the sub-and supersolution method likely fails and some other degree theoretic or variational approach can show existence.


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