Multiplicity of small or large energy solutions for Kirchhoff-Schrödinger-type equations involving the fractional p-Laplacian in R^N

We herein discuss the following elliptic equations:M(∫_R ^N∫_R ^N|u(x)-u(y)|^p/|x-y|^N+psdxdy)(-Δ)_p ^su+V(x)|u|^p-2u=Λf(x,u) in R^N, where (-Δ)_p ^s is the fractional p-Laplacian defined by (-Δ)_p ^su(x)=2lime_ε↘₀∫_R^N\Bε(x)|u(x)-u(y)|^p-2(u(x)-u(y))/|x-y|^N+psdy, 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: R₀ ⁺→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 functionMand 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.