Uniqueness And Asymptotic Stability For The Radial Solutions Of Semilinear Elliptic Equations

Abstract

Less is known of the uniqueness for the radial solutions u=ur of the problem Δu + f(u₊) = 0 in Rⁿ(n>2), u (ρ) = 0, u\'(0) = 0 besides the cases where lim<sub>r→∞u(r)=0; and for the cases based only on the evolution of the functions f(t) and ddt f(t)t. This paper proves uniqueness for the problem Da+f(u+)=0 (r>0), u(ρ) = 0, u\'(0) = 0 based on the assumption that f ∈C¹([0,∞)) and that ρ satisfies a boundedness condition. Furthermore, we prove asymptotic stability for Da+f(u+)=0 based only on the evolution of u\'(r) and u-φ(r)f(u).

Keywords: Semilinear elliptic equations, Radial solutions, uniqueness, compactness, asymptotic stability

Global Journal of Mathematical Sciences Vol. 7 (1) 2008: pp. 53-56</sub>