Uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball
This interesting paper treats the problem of uniqueness of radial solutions to the quasilinear elliptic equation \[ \text{div}(|\nabla u|^{m-2}\nabla u)+f(u)=0,\text{ in }B,\quad u>0\text{ in }B,\quad u=0\text{ on }\partial B,\tag{1} \] where \(B\) is a finite ball in \(\mathbb{R}^n\), \(n\geq 3\), and \(1<m\leq n\). Assuming that: \[ f\in C^1[0,\infty),\;f(0)=0.\tag{F1} \] \[ 0<(m-1)f(s)<sf'(s),\;\forall\;s>0,\tag{F2} \] and \(H'(s)\leq 0\), \(\forall s>0\), the authors prove that problem (1) admits at most one radial solution. Here, \[ H(s)=\overline H(s)/f(s),\;s>0,\;0,\;s=0, \] \(\overline H(s)=(n-m)sf(s)-nmF(s)\), \(F(s)=\int_0^sf(u)\;d=u\). As a consequence, they show that if \(\mu>0\), \(1\leq p<q\leq(n+2)/(n-2)\), then the problem \[ \Delta u+\mu u^p+u^q=0,\text{in }B,\;u>0\text{ in} B,\;u=0\text{ on }\partial B, \] has a unique solution when \(n\geq 6\), giving a complete answer (in the case \(n\geq 6\)), to an open problem raised by \textit{H. Brezis} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 36, 437-477 (1983; Zbl 0541.35029)]. Also, they provide partial answers to some open problems in the cases \(n= 3,4,5\). In the proofs, a new Pohozaev-type identity is established. Lastly, they show that when (F2) is not assumed, the main result is no longer valid.
scientific article; zbMATH DE number 1062193
scientific article