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 R n (n>2), u (ρ) = 0, u\'(0) = 0 besides the cases where lim r→∞ u(r)=0; and for the cases based only on the evolution of the functions f(t) and d dt f(t) t . This paper proves uniqueness for the problem D a +f(u + )=0 (r>0), u(ρ) = 0, u\'(0) = 0 based on the assumption that f ∈C 1 ([0,∞)) and that ρ satisfies a boundedness condition. Furthermore, we prove asymptotic stability for D a +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