Multiplicity of Small or Large Energy Solutions for Kirchhoff–Schrödinger-Type Equations Involving the Fractional p-Laplacian in ℝN

Abstract

We herein discuss the following elliptic equations: M ∫ R N ∫ R N | u ( x ) − u ( y ) | p | x − y | N + p s d x d y ( − Δ ) p s u + V ( x ) | u | p − 2 u = λ f ( x , u ) in R N , where ( − Δ ) p s is the fractional p-Laplacian defined by ( − Δ ) p s u ( x ) = 2 lim ε ↘ 0 ∫ R N \ B ε ( x ) | u ( x ) − u ( y ) | p − 2 ( u ( x ) − u ( y ) ) | x − y | N + p s d y , x ∈ R N . Here, B ε ( x ) : = { y ∈ R N : | x − y | < ε } , V : R N → ( 0 , ∞ ) is a continuous function and f : R N × R → R is the Carathéodory function. Furthermore, M : R 0 + → R + is a Kirchhoff-type function. This study has two aims. One is to study the existence of infinitely many large energy solutions for the above problem via the variational methods. In addition, a major point is to obtain the multiplicity results of the weak solutions for our problem under various assumptions on the Kirchhoff function M and the nonlinear term f. The other is to prove the existence of small energy solutions for our problem, in that the sequence of solutions converges to 0 in the L ∞ -norm.

Keywords

• fractional p-Laplacian
• Kirchhoff-type equations
• fountain theorem
• modified functional methods
• Moser iteration method