Uniqueness and nondegeneracy of positive solutions to elliptic equations with Robin boundary conditions

Abstract

In this paper, we study the uniqueness of positive solutions to the semilinear elliptic Robin problem $$\{\begin{array}{ll}-\Delta u=u^p,& \text{in }\Omega ,\\ u>0,& \text{in }\Omega ,\\ \frac{\partial u}{\partial \nu }+\beta u=0,& \text{on }\partial \Omega ,\end{array}$$ 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.