Uniqueness of positive solutions of nonlinear second order systems

  • Robert Dalmasso

    Equipe EDP, Grenoble, France

Abstract

In this paper we discuss the uniqueness of positive solutions of the nonlinear second order system u=g(v)–u'' = g(v), v=f(u)–v'' = f (u) in (R,R)(–R,R), u(±R)=v(±R)=0u(±R) = v(±R) = 0 where ff and gg satisfy some appropriate conditions. Our result applies, in particular, to g(v)=vg(v) = v, f(u)=upf(u) = u^p, p>1p>1, or f(u)=λu+a1up1++akupkf(u) = \lambda u + a_1u^{p1} + \cdots + a_ku^{pk} with pj>1p_j > 1, aj>0a_j > 0 for j=1,,kj = 1,\dots , k and 0λ<μ120 ≤ \lambda < \mu_1^2 where μ1=π2/4R2\mu_1 = \pi^2 / 4R^2.

Cite this article

Robert Dalmasso, Uniqueness of positive solutions of nonlinear second order systems. Rev. Mat. Iberoam. 11 (1995), no. 2, pp. 247–267

DOI 10.4171/RMI/172