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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On the nonlinear eigenvalue problem $ \Delta u+\lambda e\sp u=0$


Authors: Takashi Suzuki and Ken’ichi Nagasaki
Journal: Trans. Amer. Math. Soc. 309 (1988), 591-608
MSC: Primary 35J65; Secondary 35P30, 47H12, 47H15
MathSciNet review: 961602
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Abstract: The structure of the set $ \mathcal{C}$ of solutions of the nonlinear eigenvalue problem $ \Delta u + \lambda {e^u} = 0$ under Dirichlet condition in a simply connected bounded domain $ \Omega $ is studied. Through the idea of parametrizing the solutions $ (u,\,\lambda )$ in terms of $ s = \lambda \,\int_\Omega {{e^u}\,dx} $, some profile of $ \mathcal{C}$ is illustrated when $ \Omega $ is star-shaped. Finally, the connectivity of the branch of Weston-Moseley's large solutions to that of minimal ones is discussed.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0961602-6
PII: S 0002-9947(1988)0961602-6
Article copyright: © Copyright 1988 American Mathematical Society