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September/October 2009 Curves of equiharmonic solutions and ranges of nonlinear equations
Philip Korman
Adv. Differential Equations 14(9/10): 963-984 (September/October 2009).

Abstract

We consider the semilinear Dirichlet problem \[ \Delta u+kg(u)=\mu _1 {\varphi} _1+\ldots +\mu _n {\varphi} _n+e(x) \; \; \mbox{for $x \in U$}, \; \; u=0 \; \; \mbox{on $\partial U$}, \] where ${\varphi} _k$ is the $k$-th eigenfunction of the Laplacian on $U$ and $e(x) \perp {\varphi} _k$, $k=1, \ldots,n$. We write the solution in the form $u(x)= \sum _{i=1}^n \xi _i {\varphi} _i+U_{\xi } (x)$, with $ U_{\xi } \perp {\varphi} _k$, $k=1, \ldots,n$. Starting with $k=0$, when the problem is linear, we continue the solution in $k$ by keeping $\xi =(\xi _1, \ldots,\xi _n)$ fixed, but allowing $\mu =(\mu _1, \ldots,\mu _n)$ to vary. We then study the map $\xi \rightarrow \mu$, which provides existence and multiplicity results for the above problem.

Citation

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Philip Korman. "Curves of equiharmonic solutions and ranges of nonlinear equations." Adv. Differential Equations 14 (9/10) 963 - 984, September/October 2009.

Information

Published: September/October 2009
First available in Project Euclid: 18 December 2012

zbMATH: 1185.35095
MathSciNet: MR2548284

Subjects:
Primary: 35J60

Rights: Copyright © 2009 Khayyam Publishing, Inc.

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Vol.14 • No. 9/10 • September/October 2009
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