Uniqueness and nondegeneracy of positive solutions to elliptic equations with Robin boundary conditions
In this paper, we study the uniqueness of positive solutions to the semilinear elliptic Robin problem $$ \begin{cases} -\Delta u = u^p,&\text{in } \Omega,\\ u>0,&\text{in } \Omega,\\ \frac{\partial u}{\partial \nu} + \beta u = 0,&\text{on } \partial \Omega, \end{cases} $$ where $\beta>0$, $p$ is subcritical, and $\Omega$ is a bounded smooth domain. It is known that the uniqueness of the solution depends on the shape of the domain. Even if $\Omega$ is a ball, the problem is open for arbitrary $\beta>0$, since the method of moving planes does not work for Robin boundary conditions . By scaling arguments and a careful analysis of the linearized problem, we prove uniqueness for any $\beta>0$ provided that $p$ and $\Omega$ satisfy suitable conditions. Finally, we study the effects of concave and convex nonlinearities.