A note on a nonlinear elliptic problem with a nonlocal coefficient

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:435

Article

Naimen, Daisuke^1,2,3 

[1] Tokyo Inst Technol, Dept Math, Meguro Ku, Tokyo 1528551, Japan

[2] Japan Soc Promot Sci, Tokyo, Japan

[3] Osaka City Univ, Adv Math Inst, Sumiyoshi Ku, Osaka 5588585, Japan

关键词: Nonlocal;    Elliptic;    Bifurcation;    Variational method;    Kirchhoff type;   

DOI  :  10.1016/j.jmaa.2015.11.030

来源: Elsevier

【 摘 要 】

In this paper, we investigate a nonlocal and nonlinear elliptic problem, {-a (integral(Omega) vertical bar del(u)vertical bar(2)dx) Delta u = lambda u + u(P) in Omega, u = 0 on partial derivative Omega, (P) where N <= 3, Omega subset of R-N is a bounded domain with smooth boundary partial derivative Omega, a is a nondegenerate continuous function, p > 1 and lambda is an element of R. We show several effects of the nonlocal coefficient a on the structure of the solution set of (P). We first introduce a scaling observation and describe the solution set by using that of an associated semilinear problem. This allows us to get unbounded continua of solutions (lambda, u) of (P). A rich variety of new bifurcation and multiplicity results are observed. We also prove that the nonlocal coefficient can induce up to uncountably many solutions in a convenient way. Lastly, we give some remarks from the variational point of view. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free