Let $U$ be a open and connected subset of $\mathbb{R}^n$ with regular boundary. Consider the fallowing elliptic problem $$ \Delta u + c(x) u = u^3, U\\ \hspace{2cm}u = 0, \partial U, $$ where $c$ may change of sign.
- Supose that the bilinear form associated to the elliptic operator is coercise.
- Supode there is a solution $u \in H^1_{0}(U)$ of truncated problem $$ \Delta u + c(x) u = (u^+)^3, U.\\ $$
Under these hypotesis, I need to conclude the solution $u$ is not constant.
What I have tried: From coercivity we can conclude the solution of truncated problem is nonnegative, in other words, $u^{+} = u$. Using this fact I just can resolve the problem assuming the function $c$ is not constant. In fact, if $u$ is constant, say $u = k$, from the weak formulation $$ \int_{U} \nabla u \nabla v + c u v = \int_{U} u^{3} v, \forall v \in H^{1}_{0}. $$ we get $$ \int_{U} c v = k^2\int_{U} v, \forall v \in H^{1}_{0}. $$ In particular, $$ \int_{U} (c - k^2)v = 0, \forall v \in C^{\infty}_{0}(U). $$ This implyes $c = k^2$, what is not true if we supose the function $c$ is not constant.