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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