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Geometry of phase space and solutions of semilinear elliptic equations in a ball


Authors: Jean Dolbeault and Isabel Flores
Journal: Trans. Amer. Math. Soc. 359 (2007), 4073-4087
MSC (2000): Primary 35B33; Secondary 34C37, 34C20, 35J60
DOI: https://doi.org/10.1090/S0002-9947-07-04397-8
Published electronically: April 11, 2007
MathSciNet review: 2309176
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Abstract: We consider the problem

$\displaystyle (1)\qquad\qquad\qquad \left\{\begin{array}{ll}-\Delta u = u^{p} ... ...=0 \quad\hbox{on}\quad \partial B, \end{array}\right. \qquad\qquad\qquad\qquad $

where $ B$ denotes the unit ball in $ \mathbb{R}^N$, $ N\geq 3$, $ \lambda > 0$ and $ p>1$. Merle and Peletier showed that for $ p>\tfrac{N+2}{N-2}$ there is a unique value $ \lambda =\lambda_* >0$ such that a radial singular solution exists. This value is the only one at which an unbounded sequence of classical solutions of (1) may accumulate. Here we prove that if additionally

$\displaystyle p <\frac{N-2\sqrt{N-1}}{N-2\sqrt{N-1}-4} \quad\hbox{or} \quad N\leq 10\,, $

then for $ \lambda$ close to $ \lambda_*$, a large number of classical solutions of (1) exist. In particular infinitely many solutions are present if $ \lambda = \lambda_*$. We establish a similar assertion for the problem

$\displaystyle \left\{\begin{array}{ll}-\Delta u = \lambda\, f(u+1) \quad\hbox{i... ...ox{in}\quad B\,,\quad u=0 \quad\hbox{on}\quad \partial B\,,\end{array} \right. $

where $ f(s) = s^p + s^q$, $ 1<q<p$, and $ p$ satisfies the same condition as above.


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Additional Information

Jean Dolbeault
Affiliation: Ceremade (UMR CNRS no. 7534), Université Paris IX-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16, France
Email: dolbeaul@ceremade.dauphine.fr

Isabel Flores
Affiliation: Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 447, Chillán, Chile
Address at time of publication: Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170, Coreo 3, Santiago, Chile

DOI: https://doi.org/10.1090/S0002-9947-07-04397-8
Received by editor(s): March 24, 2004
Published electronically: April 11, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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