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Keywords:
regularizing effect; interplay; minimizer; noncoercive integral functional
regularizing effect; interplay; minimizer; noncoercive integral functional
Summary:
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 $$ \mathcal {J} (v)= \int _\Omega j(x,v,\nabla v) {\rm d}x +\int _\Omega a(x) |v|^{2} {\rm d} x -\int _\Omega fv {\rm d}x, \quad v\in W^{1,2}_{0}(\Omega ), $$ where $\Omega \subset \mathbb R^N$, $j$ 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 $$ \frac { |\xi |^{2}}{(1+|s|)^{\tau }}\leq j(x,s,\xi )\leq M|\xi |^2 $$ for almost all $x\in \Omega $, all $s\in \mathbb R$ and all $\xi \in \mathbb R^N$. We show that, even if $0<a(x)$ and $f(x)$ only belong to $L^{1}(\Omega )$, the interplay $$|f(x)|\leq 2 Qa(x) $$ implies the existence of a minimizer $u \in W_0^{1,2} (\Omega )$ which belongs to $L^{\infty }(\Omega )$.
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