The uniqueness and profile of solutions for a Matukuma-type equation on the whole space as the power of the nonlinear term tends to one are discussed. By using information on a linear eigenvalue problem and Lyapunov-Schmidt-like reduction, we obtain the uniqueness and asymptotic profile of solutions. We also consider the case of bounded domain in the same manner.
"Uniqueness and profile of solutions for a superlinear elliptic equation." Adv. Differential Equations 9 (7-8) 771 - 796, 2004.