Positive solutions for the Kirchhoff-type problem involving general critical growth - Part I: Existence theorem involving general critical growth

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:460

Article

Zhang, Huixing¹  Gu, Cong²  Yang, Chun-Ming³  Yeh, Jean²  Jiang, Juan¹ 

[1] China Univ Min & Technol, Sch Math, Xuzhou, Jiangsu, Peoples R China

[2] Texas A&M Univ, Dept Math, College Stn, TX 77843 USA

[3] Natl Taiwan Univ, Dept Math, Taipei, Taiwan

关键词: Kirchhoff-type problem;    Positive solutions;    Critical nonlinearity;   

DOI  :  10.1016/j.jmaa.2017.09.010

来源: Elsevier

【 摘 要 】

In this paper, we consider the following Kirchhoff-type problem {(a + lambda integral(R3) vertical bar del u vertical bar(2) dx + lambda b integral(R3) vertical bar u vertical bar(2) dx) (-Delta u + bu) = f(u), in R-3, u is an element of H-1 (R-3), u > 0, in R-3, where lambda >= 0 is a parameter, a, b are positive constants and f reaches the critical growth. Without the Ambrosetti-Rabinowitz condition, we prove the existence of positive solutions for the Kirchhoff-type problem with a general critical nonlinearity. We also study the asymptotics of solutions as lambda -> 0. Numerical solutions for related problems will be discussed in the second part. (C) 2017 Elsevier Inc. All rights reserved.

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