We consider a Dirichlet double phase problem with unbalanced growth. In the reaction we have the combined effects of a critical term and of a locally defined Carathéodory perturbation. Using cut-off functions and truncation techniques we bypass the critical term and deal with a coercive problem. Using this auxillary problem, we show that the original Dirichlet equation has a whole sequence of nodal (sign-changing) solutions which converge to zero in the Musielak–Orlice–Sobolev space and in L∞.

中文翻译：

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临界双相问题渐近消失的节点解

我们考虑增长不平衡的Dirichlet双阶段问题。在反应中，我们具有关键项和局部定义的Carathéodory扰动的综合作用。使用截止功能和截断技术，我们绕过了关键术语并处理了一个强制性问题。使用这个辅助问题，我们证明了原始的Dirichlet方程具有完整的节点（符号改变）解序列，它们在Musielak-Orlice-Sobolev空间和L∞中收敛为零。