Existence for $-\Delta u + u + f(u) =0$ when $f$ is a compact operator?

Question

Let $\Omega$ be a smooth domain and consider on it $$-\Delta u + u + f(u) =0 $$ with $u=0$ on $\partial\Omega$. Here $f\colon H^1(\Omega) \to H^1(\Omega)$ is a linear, compact operator, but I don't know if it is monotone and I don't have any sign information.

Is it possible to get existence for this equation (in eg. Sobolev spaces) without assuming that $\langle f(u), u \rangle$ can be "absorbed" into the lower order term $u$?

Answer 1

Since $f$ is linear, clearly $u=0$ is a solution.