标题： 退化椭圆方程的有界解与Orlicz增 Sobolev 不等式 显示英文标题

标题： Bounded solutions of degenerate elliptic equations with an Orlicz-gain Sobolev inequality

摘要： 我们考虑退化椭圆方程\[ -v^{-1}\mathrm{Div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)=f|f|^{p-2}- v^{-1}\mathrm{Div}(v|g|^{p-2}g \mathbf{t}), \quad 1<p<\infty, \]的 Dirichlet 问题解的有界性和指数可积性，假设存在形式为\[ \|\varphi\|_{L^N(v,\Omega)}\leq S_N\|\sqrt{Q} \varphi\|_{L^p(\Omega)}, \]的 Sobolev 不等式，其中$N$是形式为$N(t)=t^{\sigma p}$、$\sigma\geq 1$或者形式为$N(t)=t^p\log(e+t)^\sigma$、$\sigma>1$的 Young 函数。 在我们的结果中，我们研究了Sobolev不等式与$f$和$g$所需的正则性假设之间的相互作用，以证明解是有界的或指数可积的。 我们的结果推广了作者之前工作中证明的结果。 显示英文摘要

摘要： We consider the boundedness and exponential integrability of solutions to the Dirichlet problem for the degenerate elliptic equation \[ -v^{-1}\mathrm{Div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)=f|f|^{p-2}- v^{-1}\mathrm{Div}(v|g|^{p-2}g \mathbf{t}), \quad 1<p<\infty, \] assuming that there is a Sobolev inequality of the form \[ \|\varphi\|_{L^N(v,\Omega)}\leq S_N\|\sqrt{Q} \varphi\|_{L^p(\Omega)}, \] where $N$ is a power function of the form $N(t)=t^{\sigma p}$, $\sigma\geq 1$, or a Young function of the form $N(t)=t^p\log(e+t)^\sigma$, $\sigma>1$. In our results we study the interplay between the Sobolev inequality and the regularity assumptions needed on $f$ and $g$ to prove that the solution is bounded or is exponentially integrable. Our results generalize those previously proved in previous work by the authors.