Suppose $u\in H^1({\mathbb{R}}^n)$ has compact support and is a weak solution of the semilinear PDE $$-\Delta u+u^3=f,\ \textrm{in}\ {\mathbb{R}}^n,$$where $f\in L^2({\mathbb{R}}^n)$. Prove that $u\in H^2({\mathbb{R}}^n)$.
(Note: Hilbert space $H^k(\Omega)=W^{k,2}(\Omega)$ is the Соболев space.)
By definintion, $u$ is a weak solution implies $$\int_{{\mathbb{R}}^n}-{\delta}^{ij}D_i uD_j v dx=\int_{{\mathbb{R}}^n}(-f+u^3)v dx $$holds for any $v\in H^1({\mathbb{R}}^n)$ with compact support. But the case is different since the integration takes place on ${\mathbb{R}}^n$, which is unbounded. I can not choose a cutoff function like what is done during proving the theorem for global regularity on bounded domain. I tried to take $\Omega=B_r(0)$ and letting $r\rightarrow\infty$ but the limiting process fails when I have to take an open cover of $\partial \Omega$. That seems not the right way of working out the problem.
Any insights are appreciated. Thanks!