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:wherenonnegative 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 (), under the condition that 𝑎(𝑥) ≤ 1 with 𝑎(𝑥) → 1 as |𝑥| → ∞ and ‖𝑓‖ 𝐻 −𝑠 (ℝ 𝑁 ) is small enough (but 𝑓 ≢ 0).Further, we prove that () admits at least two positive solutions when 𝑎(𝑥) ≥ 1, 𝑎(𝑥) → 1 as |𝑥| → ∞ and ‖𝑓‖ 𝐻 −𝑠 (ℝ 𝑁 ) is small enough (but 𝑓 ≢ 0). Finally, we prove the existence of a positive solution when 𝑓 ≡ 0 under certain asymptotic behavior on the function 𝑎.