Existence of three solutions for Kirchhoff nonlocal operators of elliptic type

By Nemat Nyamoradi. 2013

49 citations recorded by the source corpus.

In this paper we prove the existence of at least three solutions to the following Kirchhoff nonlocal fractional equation: \begin{equation*} \begin{cases} M \left (\int_{\mathbb{R}^n\times \mathbb{R}^n} |u (x) - u (y)|^2 K (x - y) d x d y - \int_\Omega |u (x)|^2 d x \right) ((- \Delta)^s u - \lambda u) \\ \hspace{2cm} \in \theta (\partial j (x, u (x)) + \mu \partial k (x, u (x))), & \textrm{in}\;\; \Omega,\\ u = 0, & \textrm{in}\;\; \mathbb{R}^n \setminus \Omega, \end{cases} \end{equation*} where $(- \Delta)^s$ is the fractional Laplace operator, $s \in (0, 1)$ is a fix, $\lambda, \theta, \mu$ are real parameters and $\Omega$ an open bounded subset of $\mathbb{R}^n$, $n > 2 s$, with Lipschitz boundary. The approach is fully based on a recent three critical points theorem of Teng [K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Anal. (RWA) 14 (2013) 867-874].

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Citations in the corpus

Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem (2015, methodology): “In [30], Nyamoradi studied a class of Kirchhoff nonlocal fractional equations in a bounded domain Ω and obtained three solutions by using three critical point theorems.”
Degenerate Kirchhoff problems involving the fractional p-Laplacian without the (AR) condition (2015, result): “For some recent results in this direction, for example, see [6–9] in the non-degenerate case and [10] in the degenerate case.”
Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave–convex nonlinearities (2015, background): “Recently, in order to consider the existence of solutions to Kirchhofftype fractional problems, the authors in [35,39,41] assumed that M is an increasing function on”
Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setl (2015, methodology): “Note that in [25,39,40], in order to consider the existence of solutions to Kirchhoff type fractional problems, the authors assumed that M is an increasing function on R0 .”
Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations (2016, background): “Stationary non–degenerate Kirchhoff problems have been extensively studied in the last decades, but usually under the request that M is increasing in R0 , as in [17, 26, 29] and the reference therein.”
Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian (2015, methodology): “In [34], using a three critical points theorem for non-differential functionals, the author obtained the existence of three solutions for Kirchhoff type problems involving the nonlocal fractional Laplacian.”