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Advanced Nonlinear Studies

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Volume 18, Issue 3

Issues

On a Kirchhoff Equation in Bounded Domains

Yisheng Huang / Yuanze Wu
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  • Department of Mathematics, China University of Mining and Technology, Xuzhou 221116, P. R. China
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Published Online: 2017-12-22 | DOI: https://doi.org/10.1515/ans-2017-6042

Abstract

In this paper, we consider the following Kirchhoff equation:

{-(a+bΩ|u|2dx)Δu=λu+|u|p-2uin Ω,u=0on Ω,

where ΩN (N3) is a bounded domain with smooth boundary Ω, 2<p<2*=2NN-2 is the Sobolev exponent and a, b, λ are positive parameters. By the variational method, we obtain some existence and multiplicity results of the sign-changing solutions (including the radial sign-changing solution in the case of Ω=𝔹R) for this problem. Some further properties of these sign-changing solutions, such as the numbers of the nodal domains, the concentration behaviors as b0+, the estimates of the energy values and so on, are also obtained. Our results generalize and improve some known results in the literature. Moreover, we also obtain a uniqueness result of the radial positive solution.

Keywords: Kirchhoff-Type Equation; Positive Solution; Sign-Changing Solution; Variational Method

MSC 2010: 35B09; 35B33; 35J20

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About the article


Received: 2017-09-05

Revised: 2017-11-18

Accepted: 2017-11-25

Published Online: 2017-12-22

Published in Print: 2018-08-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11471235

Award identifier / Grant number: 11626226

Award identifier / Grant number: 11701554

Award identifier / Grant number: 11771319

Funding Source: Fundamental Research Funds for the Central Universities

Award identifier / Grant number: 2017XKQY091

Y. Huang was supported by the National Natural Science Foundation of China (11471235). Y. Wu was supported by the Natural Science Foundation of China (11626226, 11701554, 11771319) and the Fundamental Research Funds for the Central Universities (2017XKQY091).


Citation Information: Advanced Nonlinear Studies, Volume 18, Issue 3, Pages 613–648, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2017-6042.

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