Uniqueness Of Positive Solutions Of Nonlinear Elliptic Equations Radial Annular Region,

Let 0 < a < b <∞, n≥2, andΩ= {x∈Rⁿ : a < |x| < b} be an annulardomains in Rⁿ. This paper deals with the uniqueness of radial solutions of Dirichletproblemswhere f∈C¹((0,∞)×[0,∞)) and satisfies the following conditions:(A1) f(r,0)≡0 for all r > 0, and for every r > 0, there areμ1(r) >μ0(r) > 0 such thatf(r,u) < 0 whenever 0 < u <μ₀(r), f(r,u) > 0 whenever u >μ₀(r),F(r,u) < 0 whenever 0 < u <μ₁(r), F(r,u) > 0 whenever u >μ₁(r),whereF(r,u) =∫^u₀f(r,z)dz.(A2) uf_u(r,u) - f(r,u) > 0 for all r > 0 and u > 0.(A3) f_r(r,u)≤0 and 2F_r(r,u)≤uf_r(r,u) for all r > 0 and u > 0.(A4) The functiong(r,u) = (2f(r,u)+rf_r(r,u))/(uf_u(r,u)-f(r,u))is nondecreasing with respect to u for fixed r > 0, and nonincreasing with respectto r for fixed u > 0.(A5) The functionG(r,u) = (2F(r,u)+rF_r(r,u))/(uf(r,u)-2F(r,u))is nondecreasing with respect to u for fixed r > 0, and nonincreasing with respectto r for fixed u > 0.(A6) For c > 0,η>μ1(c), r∈(0,c) and u∈(0,η),k(r,u,c,η) =g(c,η)[uf_u(r,u)- f(r,u)]u + [2F_r(r,u)- uf_r(r,u)]r-n[uf(r,u)- 2F(r,u)]- 2nF(c,η)≤0.