Existence of Solutions for Schr?dinger–Kirchhoff Type Problems Involving Nonlocal Elliptic Operators

摘要

The aim of this paper is to establish the existence of nonnegative solutions for a class of Schr?dinger–Kirchhoff type problems driven by nonlocal integro-differential operators, that is,M?R2N|u(x)?u(y)|pK(x?y)dxdy,∫RNV(x)|u|pdxLKu+V(x)|u|p?2u=G?R2N|u(x)?u(y)|pK(x?y)dxdy,∫RNV(x)|u|pdxf(x,u)+h(x)????in?RN,$$egin{align*}&Mleft(mathop{iint_{mathbb{R}^{2N}}}|u(x)-u(y)|^pK(x-y)dxdy,int_{mathbb{R}^N}V(x)|u|^pdxight)&kern10pt left(mathcal{L}_Ku+V(x)|u|^{p-2}uight)&=Gleft(mathop{iint_{mathbb{R}^{2N}}}|u(x)-u(y)|^pK(x-y)dxdy,int_{mathbb{R}^N}V(x)|u|^pdxight)onumber&kern11pt f(x,u)+h(x) {m in} mathbb{R}^N,end{align*}$$where LK$mathcal{L}_K$ is a nonlocal integro-differential operator with singular kernel K:RN?{0}→(0,∞)$K:mathbb{R}^N,ackslash,{0}ightarrow(0,infty)$, M,G$M,G$ are two nonnegative continuous functions on (0,∞)×(0,∞)$(0,infty)imes(0,infty)$, V∈C(RN,R+)$Vin C(mathbb{R}^N,mathbb{R}^+)$, h:RN→(0,∞)$h:mathbb{R}^Nightarrow (0,infty)$ is a measurable function and f:RN×R→R$f:mathbb{R}^Nimesmathbb{R}ightarrowmathbb{R}$ is a Carathéodory function. Employing several nonvariational techniques, we prove various results of existence of nonnegative solutions. The main feature of this paper is that the Kirchhoff function M$M$ can be zero at zero and the problem is not variational in nature.