Bifurcation results for a class of prescribed mean curvature equations in bounded domains. (June 2018)

• Record Type: Journal Article
• Authors: Pan, Hongjing; Xing, Ruixiang
• Is Part Of: Nonlinear analysis. Volume 171(2018)
• Page Start: 21
• Page End: 31
• Publication Date: 2018-06
• Subjects: 35J93 -- 47J15 -- 35B32 -- 34B15; Prescribed mean curvature equation -- Direction of bifurcation -- Nonlinear eigenvalue problem -- Transcritical bifurcation -- Pitchfork bifurcation -- Exponential nonlinearity -- Logistic nonlinearity -- Pendulum -- Perturbed Gelfand problem; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.01.010
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Bifurcation for prescribed mean curvature equations in one dimension has been intensively investigated in recent years, and a striking phenomenon discovered is that the length of the interval may affect the shapes of bifurcation curves. However, to our best knowledge, no such results are known in higher dimensions. In this paper, we study a class of prescribed mean curvature equations in bounded domains of R N with general nonlinearities f satisfying f ( 0 ) = 0 and f ′ ( 0 ) > 0 . We establish some formulas for directions of bifurcation at simple eigenvalues, which lead to a sufficient and necessary condition to ensure that the directions depend on the size of the domain. In contrast, this phenomenon does not occur for the semilinear case. Some interesting examples, including logistic and perturbed exponential nonlinearities, are also investigated.