I recently encountered two papers discussing elliptic PDEs and variational methods. The first paper claims that according to regularity theory, the solution to $-\Delta u = ug(u^2)$ in $\mathbb{R}^2$ belongs to $W_{\text{loc}}^{2,q}(\mathbb{R}^2)$ for any $q > 2$, provided $g(s)\leq Ce^{Cs}$ for some positive constant $C$.
The second paper suggests that if $g(s)\leq Ce^{Cs^2}$ for some positive constant $C$, then the variational functional $$J(u)=\int_{\Omega}\Big{[} \frac{1}{2}|\nabla u|^2+ \chi_{\{u>1\}}(x)-G((u-1)_+) \Big{]}dx$$ is well defined on the Sobolev space $H_{0}^1(\Omega)$, where $\Omega \subset \mathbb{R}^2$ is a smooth bounded domain, $(u-1)_+$ is the positive part of $u-1$, and $G$ is the primitive of $g$.
I am somewhat perplexed by these claims. It seems to me that they rely on the Moser-Trudinger inequality. However, the Moser-Trudinger inequality typically requires that the constant $C$ in $e^{Cu^2}$ satisfies $C < 4\pi$, but there is no such restriction on $C$ in these claims.
(Page 6 of 14): The first paper
(Page 2 of 32): The second paper
Could someone clarify how these claims are justified given the growth rates of $g(s)$? Are there alternative techniques employed in these papers that circumvent the typical constraints of the Moser-Trudinger inequality? Any insights or references would be greatly appreciated.
