Uniqueness of positive solutions of \(\Delta u+f(u)=0\) in \(\mathbb{R}^N\), \(N\geqq 3\)
The authors prove the uniqueness of nonnegative solutions of \[ \Delta u+ f(u)= 0,\quad x\in\mathbb{R}^N,\quad N\geq 3, \] which decay to zero at infinity, when the nonlinearity \(f\) has superlinear growth. In particular, \(f: [0,\infty)\to \mathbb{R}\) satisfies: i) \(f(0)= 0\) and there exists \(u_0> 0\) such that \(f(u)>0\) for \(u> u_0\) and \(f(u)\leq 0\), \(f(u)\not\equiv 0\), for \(u\in [0, u_0)\), ii) \(f\) is continuous on \([0,\infty)\) and locally Lipschitz continuous on \([u_0, \infty)\), iii) \(f(u)\leq f'(u)(u- u_0)\) for all \(u\in D\), and iv) \(S(u)= uf'(u)/f(u)\) is monotone decreasing on \(D\), for \(D\) the subset of \([u_0,\infty)\) where \(f\) is differentiable. The main result is accomplished by using \textit{C. V. Coffman's} function \(\varphi\) [Arch. Ration. Mech. Anal. 46, 81-95 (1972; Zbl 0249.35029)] defined in terms of the solution of an associated initial value problem and following a monotone separation theorem of \textit{L. A. Peletier} and \textit{J. Serrin} [J. Differ. Equations 61, 380-397 (1986; Zbl 0577.35035)].
scientific article; zbMATH DE number 1176733