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Let $A$ be a CM-finite Artin algebra with a Gorenstein-Auslander generator $E$, $M$ be a Gorenstein projective $A$-module and $B={\mathrm{End}}_{A}M$. We give an upper bound for the finitistic dimension of $B$ in terms of homological data of $M$. Furthermore, if $A$ is $n$-Gorenstein for $2\le n<\infty$, then we show the global dimension of $B$ is less than or equal to $n$ plus the $B$-projective dimension of ${\mathrm{Hom}}_A(M,E).$ As an application, the global dimension of ${\mathrm{End}}_{A}E$ is less than or equal to $n$.

We are interested in regularizing effect of the interplay between the coefficient of zero order term and the datum in some noncoercive integral functionals of the type $$𝒥\left(v\right)={\int }_{\Omega }j(x,v,\nabla v)\mathrm{d}x+{\int }_{\Omega }a\left(x\right){\left|v\right|}^2\mathrm{d}x-{\int }_{\Omega }fv\mathrm{d}x,v\in W_0^{1,2}\left(\Omega \right),$$ where $\Omega \subset {ℝ}^N$, $j$ is a Carathéodory function such that $\xi \mapsto j(x,s,\xi )$ is convex, and there exist constants $0\le \tau <1$ and $M>0$ such that $$\frac{{\left|\xi \right|}^2}{{(1+|s\left|\right)}^{\tau }}\le j(x,s,\xi )\le M{\left|\xi \right|}^2$$ for almost all $x\in \Omega$, all $s\in ℝ$ and all $\xi \in {ℝ}^N$. We show that, even if $0<a\left(x\right)$ and $f\left(x\right)$ only belong to $L^1\left(\Omega \right)$, the interplay $$\left|f\right(x\left)\right|\le 2Qa\left(x\right)$$ implies the existence of a minimizer $u\in W_0^{1,2}\left(\Omega \right)$ which belongs to $L^{\infty }\left(\Omega \right)$.