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

Abstract

AbstractWe study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations: where , , , and is a nonnegative functional, that is, whenever u is a nonnegative function in . We establish Palais–Smale decomposition of the functional associated with the above equation. Using the decomposition, we establish existence of three positive solutions to (), under the condition that with as and is small enough (but ). Further, we prove that () admits at least two positive solutions when , as and is small enough (but ). Finally, we prove the existence of a positive solution when under certain asymptotic behavior on the function a.