Existence and multiplicity of positive solutions of certain nonlocal scalar field equations

Abstract

We study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations:.{(-Delta)(s) u +u =a (x)|u|(p-1)u +f (x) in R-N ( p), u is an element of H-s(R-N) where s is an element of (0,1), N> 2s, 1(s)(H) >= 0 whenever u is a nonnegative function in H-s (R-N). We establish Palais-Smale decomposition of the functional associated with the above equation. Using the decomposition, we establish existence of three positive solutions to (p), under the condition that a(x) <= 1 with a(x) -> 1 as |x|->infinity and parallel to f parallel to H (-s) (R-N) is small enough (but f not equivalent to 0). Further, we prove that (p) admits at least two positive solutions when a(x) >= 1, a(x)-> 1 as | x |->infinity and parallel to f parallel to H-s (R-N) is small enough (but f not equivalent to 0). Finally, we prove the existence of a positive solution when. f equivalent to 0 under certain asymptotic behavior on the function..