Uniqueness of positive radial solutions for \(n\)-Laplacian Dirichlet problems (Q2708160)

The author studies the uniqueness of positive radial solutions for the boundary value problem NEWLINE\[NEWLINE\begin{gathered}\text{div}(|\nabla u|^{m-2}\nabla u)+ f(u)= 0\quad\text{in }B,\\ u> 0\quad\text{in }B,\quad u=0\quad\text{on }\partial B,\end{gathered}\tag{1}NEWLINE\]NEWLINE where \(B\) is a finite ball in \(\mathbb{R}^n\), as well as of the associated problem NEWLINE\[NEWLINE\text{div}(|\nabla u|^{m- 2}\nabla u)+ f(u)= 0,\quad u>0\quad\text{in }\mathbb{R}^n.\tag{2}NEWLINE\]NEWLINE The following theorem is proved: Let \(n\leq m\). If \(f\) satisfiesNEWLINENEWLINENEWLINE(H1) \(f\in C^1 0,\infty)\), \(f(0)= 0\), \(f(u)> 0\) on \((0,\infty)\),NEWLINENEWLINENEWLINEthen (2) admits no radial solutions in \(\mathbb{R}^n\). Moreover, if \(f\) also satisfiesNEWLINENEWLINENEWLINE(H2) \(f(u)/u^{m-1}\) is strictly increasing over \((0,\infty)\),NEWLINENEWLINENEWLINE(H3) \(\int^u_0 f(s) ds/f(u)\) is increasing over \((0,\infty)\),NEWLINENEWLINENEWLINEthen the Dirichlet problem (1) has at most one radial solution.