Global integrability for solutions to some anisotropic problem with nonstandard growth

Abstract

AbstractThis paper deals with the problem\displaystyle u\in{\cal K}_{u_{*},\psi}(\Omega),\displaystyle\forall v\in{\cal K}_{u_{*},\psi}(\Omega):\int_{\Omega}\sum_{i=1}% ^{n}[a_{i}(x,Du)-f^{i}]D_{i}(u-v)\,dx\leqslant\int_{\Omega}f(u-v)\,dx,where\left\{\begin{aligned} &\displaystyle{\cal K}_{u_{*},\psi}(\Omega)=\biggl{\{}v% \in u_{*}+W_{0}^{1,(p_{i})}(\Omega):\sum_{i=1}^{n}a_{i}(x,Du)D_{i}v\in L^{1}(% \Omega)\text{ and }v\geqslant\psi,\text{ a.e. }\Omega\biggr{\}},\\ &\displaystyle u_{*}\in W^{1,(p_{i})}(\Omega),\quad\theta=\max\{u_{*},\psi\}% \in u_{*}+W_{0}^{1,(p_{i})}(\Omega),\\ &\displaystyle f\in L^{(\bar{p}^{*})^{\prime}}(\Omega),\quad f^{i}\in L^{p_{i}% ^{\prime}}(\Omega),\,i=1,\dots,n,\end{aligned}\right.and the Carathéodory functions {a_{i}:\Omega\times{\mathbb{R}}^{n}\to{\mathbb{R}}}, {i=1,\dots,n}, satisfy some coercivity condition. We assume that the function {\theta=\max\{u_{*},\psi\}} makes {a_{i}(x,D\theta)} to be more integrable than {L^{p_{i}^{\prime}}(\Omega)}, {i=1,\dots,n}, and then we prove that the solution u enjoys higher integrability.