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Positive Solutions of a Nonlocal and Nonvariational Elliptic Problem

Authors
  • Liu, Lingjun1
  • Shi, Feilin2
  • 1 Chinese Academy of Sciences, Beijing, 100190, China , Beijing (China)
  • 2 Hunan Normal University, Changsha, 410081, China , Changsha (China)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Jun 29, 2021
Volume
41
Issue
5
Pages
1764–1776
Identifiers
DOI: 10.1007/s10473-021-0522-5
Source
Springer Nature
Keywords
Disciplines
  • Article
License
Yellow

Abstract

In this paper, we will study the nonlocal and nonvariational elliptic problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\matrix{{ - (1 + a\|u\|_q^{\alpha q})\Delta u = |u{|^{p - 1}}u + h(x,u,\nabla u)\,{\rm{in}}\,\,\,\Omega ,} \hfill \cr {u = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,on\,\,\,\partial \Omega ,} \hfill \cr } } \right.$$\end{document} where a > 0, α > 0, 1 < q < 2*, p ∈ (0, 2* − 1) {1} and Ω is a bounded smooth domain in ℝN (N ≥ 2). Under suitable assumptions about h(x, u, ∇u), we obtain a priori estimates of positive solutions for the problem (0.1). Furthermore, we establish the existence of positive solutions by making use of these estimates and of the method of continuity.

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