Affiliation:
1. 1College of Science, Civil Aviation University of China, Tianjin, 300300, P.R. China
Abstract
Abstract:The aim of this paper is to establish the existence of nonnegative solutions for a class of Schrödinger–Kirchhoff type problems driven by nonlocal integro-differential operators, that is,
$$\begin{align*}&M\left(\mathop{\iint_{\mathbb{R}^{2N}}}|u(x)-u(y)|^pK(x-y)dxdy,\int_{\mathbb{R}^N}V(x)|u|^pdx\right)\\&\kern10pt \left(\mathcal{L}_Ku+V(x)|u|^{p-2}u\right)\\&=G\left(\mathop{\iint_{\mathbb{R}^{2N}}}|u(x)-u(y)|^pK(x-y)dxdy,\int_{\mathbb{R}^N}V(x)|u|^pdx\right)\nonumber\\&\kern11pt f(x,u)+h(x)\ \ \ \ {\rm in}\ \mathbb{R}^N,\end{align*}$$where $\mathcal{L}_K$ is a nonlocal integro-differential operator with singular kernel $K:\mathbb{R}^N\,\backslash\,\{0\}\rightarrow(0,\infty)$, $M,G$ are two nonnegative continuous functions on $(0,\infty)\times(0,\infty)$, $V\in C(\mathbb{R}^N,\mathbb{R}^+)$, $h:\mathbb{R}^N\rightarrow (0,\infty)$ is a measurable function and $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ 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 $M$ can be zero at zero and the problem is not variational in nature.
Subject
Applied Mathematics,General Physics and Astronomy,Mechanics of Materials,Engineering (miscellaneous),Modeling and Simulation,Computational Mechanics,Statistical and Nonlinear Physics
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