Differential Equations and Computational Simulations III

Electron. J. Diff. Eqns., Conf. 01, 1997, pp. 119, 127.

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A global solution curve for a class of semilinear equations

Philip Korman

**Abstract:**

We use bifurcation theory to give a simple proof of existence and
uniqueness of a positive solution for the problem

for
, u = 0 on |x| = 1,

where
,
for any positive integer n, and real
,
.
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

for ,
u = 0 on |x| = 1

is one, for any
.
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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