Hi I am interested in what the exact purpose is of growth conditions associated with solving partial differential equations. For example the following pde: $$\text{div}(a(x,u,\nabla u)) + c(c,u,\nabla u) = g$$ where examples of some of the growth conditions associated with it are $$|a(x,r,s)| \leq \gamma(x) + C|r|^{\frac{(p^{*}-\epsilon)}{p'}} + C|s|^{p-1}$$ and
$$|c(x,r,s)| \leq \gamma(x) + C|r|^{p^{*}-\epsilon -1} + C|s|^{\frac{p}{p^{*'}}}$$
Can someone with some knowledge of pde's give me an idea of why growth conditions are necessary.
The PDE of this form is typically posed in some Sobolev space $W^{1,p}$. At a minimum, we need the weak formulation $$\int (-a(x,u,\nabla u)\nabla \phi + c(x,u,\nabla u) \phi - g\phi)=0 \tag{1}$$ to make sense for $\phi\in C^\infty_0$. But often, to actually use $(1)$ one needs $\phi$ to be somehow based on $u$. So we'd like to allow $\phi$ in a Sobolev space too, probably the same $W^{1,p}$. Then $(1)$ requires $ a(x,u,\nabla u)\in L^{p'}$ and $c(c,u,\nabla u)\in L^{(p^*)'}$. Since $u\in L^{p^*}$ and $\nabla u\in L^p$, we need $a=O(|u|^{p^*/p'})$ and $a=O(|\nabla u|^{p/p'})$ (where $p/p'=p-1$) and so forth. That some $\epsilon$ appears in the exponents probably has to do with the specific method employed by the authors of the paper you are reading. But the above gives the general idea of what motivates those assumptions.