Positive solutions of the Robin problem for semilinear elliptic equations on annuli (Q993417)

Summary: Let \(n\geq 3\) and \(\Omega_R= \{x\in\mathbb R^n\); \(R<|x|<1\}\). We consider the following Robin problem: \[ \begin{cases} -\Delta u=f(u), &x\in\Omega_R,\\ u>0, &x\in\Omega_R,\\ \dfrac{\partial u}{\partial \nu}+\beta u=0 &x\in\partial\Omega_R, \end{cases} \] where \(\beta\) is a positive parameter and \(\nu\) is the unit outward vector normal to \(\partial\Omega_R\). Under some assumptions, we prove that the above problem has at most one solution when \(\beta\) is small enough, and under some other assumptions, the above problem has at least \(k\) nonradial solutions when \(\beta\) is large enough.