Suppose that $B_1 \subset \mathbb{R}^3$ and $P(u)$ is a polynomial with degree 3. If $u \in W^{1,2}(B_1)$ is a weak solution of $$\Delta u = P(u) \text{ in } B_1,$$ then can we obtain the smoothness of the solution?
I found that the theories in Trudinger's book can not be applied since the integrability of $P(u)$ is not good enough. And If the degree of $P$ is higher, I found there may not exist a smooth solution.
Is that true if I replace $P(u)$ by a smooth function $g(u)$ such that $\lim_{x \to \infty}\dfrac{g}{u^3}<\infty$? May I have a reference of it? Thank you!