Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 117, pp. 1-14.

Title: Direct and inverse bifurcation problems for non-autonomous 
       logistic equations
        
Author: Tetsutaro Shibata (Hiroshima Univ., Higashi-Hiroshima, Japan)

Abstract:
 We consider the semilinear eigenvalue problem
 $$\displaylines{
 -u''(t) + k(t)u(t)^p = \lambda u(t), \quad u(t) > 0, \quad
 t \in I := (-1/2, 1/2), \cr
 u(-1/2) = u(1/2) = 0,
 }$$
 where p > 1 is a constant,
 and $\lambda > 0$ is a parameter. We propose a new
 inverse bifurcation problem.  Assume that k(t) is an  unknown
 function. Then can we determine k(t) from the asymptotic behavior of
 the bifurcation curve?
 The purpose of this paper is to answer this question affirmatively.
 The key ingredient is the precise asymptotic formula for
 the $L^q$-bifurcation curve $\lambda = \lambda(q,\alpha)$
 as $\alpha \to \infty$ ($1 \le q < \infty$),
 where $\alpha := \| k^{1/(p-1)}u_\lambda\|_q$.

Submitted  December 28, 2012. Published May 10, 2013.
Math Subject Classifications: 34C23, 34B15.
Key Words: Inverse bifurcation problem; non-autonomous equation; 
           logistic equations.