Electronic Journal of Differential Equations,
Conference 01 (1998), pp. 119, 127.

Title: A Global Solution Curve for a Class of Semilinear Equations 


Authors: Philip Korman (Univ. of Cincinnati, Cincinnati, Ohio, USA)
 
Abstract: 
We use bifurcation theory to give a simple proof of existence and
uniqueness of a positive solution for the problem
$$
\Delta u - \lambda u+u^p = 0 \quad
\mbox{for } |x| < 1, \quad u = 0 \quad \mbox{on } |x| = 1,
$$
where $x \in {\mathbb R}^n$, for any integer $n \geq 1$, and real 
$1<p < (n+2)/(n-2)$, $\lambda \geq 0$.
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
$$
\Delta u +u^p = 0 \quad
\mbox{for } |x| < 1, \quad u = 0 \quad \mbox{on } |x| = 1
$$
is one, for any $1<p < (n+2)/(n-2)$.


Published November 12, 1998.
Math Subject Classifications: 35J60.
Key Words: Uniqueness of positive solution; Morse index.