JournalszaaVol. 18, No. 3pp. 517–523

On Uniqueness Conditions for Decreasing Solutions of Semilinear Elliptic Equations

  • Tadie

    Copenhagen University, Denmark
On Uniqueness Conditions for Decreasing Solutions of Semilinear Elliptic Equations cover
Download PDF

Abstract

For fC((0,))C1((0,))f\in C((0, \infty)) \cap C^1 ((0, \infty)) and b>0b > 0, existence and uniqueness of radial solutions u=u(r)u = u(r) of the problem Δu+f(u+)=0\Delta u + f(u_+) = 0 in Rn(n>2),u(0)=b\mathbb R^n (n > 2), u(0) = b and u(0)=0u’(0) = 0 are well known. The uniqueness for the above problem with boundary conditions u(R)=0u(R) = 0 and u(0)=0u’(0) = 0 is less known beside the cases where limru(r)=0lim_{r \to \infty} u(r) = 0. It is our goal to give some sufficient conditions for the uniqueness of the solutions of the problem Dαu+f(u+)=0(r>0),u(p)=0D_{\alpha}u + f(u_+) = 0 (r > 0), u(p) = 0 and u(0)=0u’(0) = 0 based only on the evolution of the functions f(t)f(t) and ddtf(t)t\frac{d}{dt} \frac{f(t)}{t}.

Cite this article

Tadie, On Uniqueness Conditions for Decreasing Solutions of Semilinear Elliptic Equations. Z. Anal. Anwend. 18 (1999), no. 3, pp. 517–523

DOI 10.4171/ZAA/895