help me in the following situation: If we take the equation $$-\Delta u + u^4 = 0 \ \ \mbox{in} \ \ \mathbb{R}^2,$$ with $u \in C^2(\mathbb{R}^2)$, then it is easy to see that $u \equiv 0$ is a trivial solution to the problem. Hence, the following question arises: is it possible to find another non-constant $u$ function that satisfies the previous problem?
This caused me doubt because it is well known that if we assign boundary conditions, then we can guarantee the uniqueness of problems similar to the one I proposed.