Let $m\in \mathbb{N}\setminus \{1\}$ and suppose $(M,g)$ denotes a compact smooth Riemannian manifold with smooth boundary and consider the semi-linear equation $$-\Delta_g u+q(x)u + a(x)u^m=0\quad \text{on}\quad M,$$ subject to $u|_{\partial M}=f$. Here $q$ and $a$ are smooth and $a>0$ and we additionally impose that $0$ is not a Dirichlet eigenvalue for $-\Delta_g+q$.
What can be said about existence, uniqueness and smoothness of solutions? (I know when $m$ is odd this follows quite easily and classically but what about the case $m$ even?)
Thanks,
