On uniqueness conditions for decreasing solutions of semilinear elliptic equations (Q1806176)

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