MR2154380 Korman, Philip Philip KORMAN Existence and Uniqueness of Solutions for a Class of Non-Autonomous Dirichlet Problems Funkcialaj Ekvacioj. Serio Internacia 48 2005 99--111 http://fe.math.kobe-u.ac.jp/FE/FullPapers/48-1/48_99.pdf http://www.ams.org/mathscinet-getitem?mr=MR2154380 We prove that the semilinear Dirichlet problem for a Laplace equation on a unit ball, involving the nonlinearity $f(r,u)=-a(r)u+b(r)u^p$, with a subcritical $p$, has a unique positive solution, provided $a(r)$ is positive, increasing and convex, while $b(r)$ is positive, decreasing and concave. Moreover, we prove that this solution is non-degenerate. We also present a uniqueness result in case $a(r)$ is negative. Existence and uniqueness of solutions. 35J60. 48-99 2005 Existence and Uniqueness of Solutions for a Class of Non-Autonomous Dirichlet Problems KORMAN, Philip Philip KORMAN 1 Chang, K. C. Infinite-dimensional Morse theory and multiple solution problems Progress in Nonlinear Differential Equations and their Applications, 6. Birkhäuser Boston, Inc., Boston, MA 1993 MR1196690
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