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Semilinear Neumann boundary value problems on a rectangle


Author: Junping Shi
Journal: Trans. Amer. Math. Soc. 354 (2002), 3117-3154
MSC (2000): Primary 35J25, 35B32; Secondary 35J60, 34C11
DOI: https://doi.org/10.1090/S0002-9947-02-03007-6
Published electronically: April 2, 2002
MathSciNet review: 1897394
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Abstract: We consider a semilinear elliptic equation

\begin{displaymath}\Delta u+\lambda f(u)=0, \;\; \mathbf{x}\in \Omega,\;\; \frac{\partial u}{\partial n }=0, \;\; {\mathbf x}\in \partial \Omega, \end{displaymath}

where $\Omega$ is a rectangle $(0,a)\times(0,b)$ in $\mathbf{R}^2$. For balanced and unbalanced $f$, we obtain partial descriptions of global bifurcation diagrams in $(\lambda,u)$ space. In particular, we rigorously prove the existence of secondary bifurcation branches from the semi-trivial solutions, which is called dimension-breaking bifurcation. We also study the asymptotic behavior of the monotone solutions when $\lambda\to\infty$. The results can be applied to the Allen-Cahn equation and some equations arising from mathematical biology.


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Additional Information

Junping Shi
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187, and Department of Mathematics, Harbin Normal University, Harbin, Heilongjiang, P. R. China 150080
Email: shij@math.wm.edu

DOI: https://doi.org/10.1090/S0002-9947-02-03007-6
Keywords: Semilinear elliptic equations, secondary bifurcations, global bifurcation diagrams, asymptotic behavior of solutions
Received by editor(s): April 17, 2001
Published electronically: April 2, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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