Semilinear Neumann boundary value problems on a rectangle
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- by Junping Shi
- Trans. Amer. Math. Soc. 354 (2002), 3117-3154
- DOI: https://doi.org/10.1090/S0002-9947-02-03007-6
- Published electronically: April 2, 2002
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Abstract:
We consider a semilinear elliptic equation \begin{equation*} \Delta u+\lambda f(u)=0, \;\; \mathbf {x}\in \Omega ,\;\; \frac {\partial u}{\partial n }=0, \;\; \mathbf {x}\in \partial \Omega , \end{equation*} 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.References
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Bibliographic 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
- MR Author ID: 616436
- ORCID: 0000-0003-2521-9378
- Email: shij@math.wm.edu
- Received by editor(s): April 17, 2001
- Published electronically: April 2, 2002
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1897394