Uniqueness of solution for Dirichlet and Neumann problem

Question

How to prove the uniqueness of the Dirichlet and Neumann problems for the equation $$ \Delta u - ku=0$$ in a bounded domain $D \subseteq \mathbb{R}^2$, where $K$ is a positive constant? What happens if $K<0$?

Thanks.

Answer 1

If you assume you have two solutions $u_1, u_2$ consider their difference $u=u_1-u_2$. If not identically zero, it attains either a positive max. or a negative min at some $x_0$ inside $D$. Suppose it is a positive max. Then $\Delta u(x_0)\le 0$ and $-ku(x_0)<0$ so the equation cannot be satisfied at $x_0$. You get a similar contradiction at a negative minimum of $u$. For $k<0$ you may have solutions if $k$ is in the point spectrum of $\Delta$ with homogeneous Dirichlet boundary conditions.