Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems

Abstract

We apply the “monotone separation of graphs” technique of L.A. Peletier and J. Serrin [L.A. Peletier, J. Serrin, Uniqueness of positive solutions of semilinear equations in R n , Arch. Ration. Mech. Anal. 81 (2) (1983) 181–197; L.A. Peletier, J. Serrin, Uniqueness of nonnegative solutions of semilinear equations in R n , J. Differential Equations 61 (3) (1986) 380–397], as developed further by L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179–202], to the question of exact multiplicity of positive solutions for a class of semilinear equations on a unit ball in R n . We also observe that using P. Pucci and J. Serrin [P. Pucci, J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J. 47 (2) (1998) 501–528] improvement of a certain identity of L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179–202] produces a short proof of L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179–202] result on the uniqueness of positive solution of ( 1 < p , q < n + 2 n − 2 ) Δ u + u p + u q = 0 for | x | < 1 , u = 0 when | x | = 1 .