LAUSR

We herein discuss the following elliptic equations:M ( ∫ RN ∫ RN |u(x)−u(y)|p |x−y|N+ps dx dy ) (−∆)pu+ V(x)|u|p−2u = λ f (x, u) inRN, where (−∆)p is the fractional p-Laplacian defined by (−∆)pu(x) = 2 limε↘0 ∫ RN\Bε(x) |u(x)−u(y)|p−2(u(x)−u(y)) |x−y|N+ps dy, x ∈ R N . Here, Bε(x) := {y ∈ RN : |x − y| < ε}, V : RN → (0, ∞) is a continuous function and f : RN × 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 functionM 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.