Advances in Differential Equations

Bifurcation for nonlinear elliptic boundary value problems. III.

Kazuaki Taira and Kenichiro Umezu

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This paper is devoted to global static bifurcation theory for a class of degenerateboundary value problems for nonlinear second-order elliptic differential operators which includes as particular cases the Dirichlet and Neumann problems. In the previous paper [13] we treated the asymptotic linear case, for example, such nonlinear terms as $u^p$, $p > 1$, near $u = 0$ but $u + 1/u$ near $u = +\infty$. The purpose of this paper is to study the asymptotic nonlinear case, for example, such nonlinear terms as $u^p$ also near $u = +\infty$. First we prove a general existence and uniqueness theorem of positive solutions for our nonlinear boundary value problems, by using the super-subsolution method, and then we study in great detail the asymptotic nonlinear case.

Article information

Adv. Differential Equations, Volume 1, Number 4 (1996), 709-727.

First available in Project Euclid: 25 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 35J25: Boundary value problems for second-order elliptic equations


Taira, Kazuaki; Umezu, Kenichiro. Bifurcation for nonlinear elliptic boundary value problems. III. Adv. Differential Equations 1 (1996), no. 4, 709--727.

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