Uniqueness of solutions of the elliptic equation $\Delta u = u^5 + 2 u^3 + 3 u$
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Assume that $u \in C^2(\bar \Omega) \cap H_0^1 (\Omega)$ be a strong solution to
\[ \Delta u = u^5 + 2 u^3 + 3 u \qquad \text{in} \Omega, \]
with $u=0$ on $\partial \Omega$. Show that $u \equiv 0$ is the only solution.
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