Differential Equations and Computational Simulations III
Electron. J. Diff. Eqns., Conf. 01, 1997, pp. 119, 127.

A global solution curve for a class of semilinear equations

Philip Korman

We use bifurcation theory to give a simple proof of existence and uniqueness of a positive solution for the problem
$$\Delta u - \lambda u+u^p = 0$$ for $|x|$ less than 1, u = 0 on |x| = 1,
where $x \in {\Bbb R}^n$, for any positive integer n, and real 1 less than p less than (n+2)/(n-2), $\lambda \geq 0$. Moreover, we show that all solutions lie on a unique smooth curve of solutions, and all solutions are non-singular. In the process we prove the following assertion, which appears to be of independent interest: the Morse index of the positive solution of
$$ \Delta u +u^p = 0$$ for $|x|$ less than 1, u = 0 on |x| = 1
is one, for any 1 less than p less than (n+2)/(n-2).

Published November 12, 1998.
Mathematics Subject Classifications: 35J60.
Key words and phrases: Uniqueness of positive solution, Morse index.

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Philip Korman
Institute for Dynamics and
Department of Mathematical Sciences
University of Cincinnati
Cincinnati Ohio 45221-0025 USA
E-mail address: kormanp@math.uc.edu

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