# Compactness of solutions of elliptic equation

Consider the following nonlinear elliptic equation $$-\triangle u + u + u^3 = g, \quad x \in R^3.$$ If $g \in L^2(R^3)$, then the set $Q$ of solutions of above equation is bounded in $H^2(R^3)$, and it is compact in $L^2(R^3)$.

Now we introduce the locally uniform spaces $L^2_U$ consisting of functions satisfying $$\|\varphi\|_{L^2_U} := \sup_{y\in R^3} \left(\int_{B_y(1)}|u|^2dx\right)^{1/2}< \infty$$ where $B_{y}(1) = \{x\in R^3: |x - y| \leq 1\}$.

If $g \in L^2_U$, then the set $Q$ of solutions of above equation is still bounded in $H^2_U$(You understand the definition!). My question is that, it is still compact in $L^2_U$?

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