| Summary: | 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 $alpha o infty$ ($1 le q < infty$), where $alpha := | k^{1/(p-1)}u_lambda|_q$.
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