Multiplicity result for non-homogeneous fractional Schrodinger--Kirchhoff-type equations in ℝn

Abstract

AbstractIn this paper we consider the existence of multiple solutions for the non-homogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff typeM\Bigg{(}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(z)|^{p}}{|x-{% z}|^{n+ps}}\,dz\,dx\Bigg{)}(-\Delta)_{p}^{s}u+V(x)|u|^{p-2}u=f(x,u)+g(x)in {\mathbb{R}^{n}}, where (-Δ)_{p}^{s} is the fractional p-Laplacian operator with 0¡s¡1¡p¡\infty, ps¡n, f : \mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R} is a continuous function, V : \mathbb{R}^{n}\to\mathbb{R}^{+} is a potential function and g : \mathbb{R}^{n}\to\mathbb{R} is a perturbation term. Assuming that the potential V is bounded from bellow, that f(x,t) satisfies the Ambrosetti–Rabinowitz condition and some other reasonable hypotheses, and that g(x) is sufficiently small in L^{p^{\prime}}(\mathbb{R}^{n}), we obtain some new criterion to guarantee that the equation above has at least two non-trivial solutions.