The behavior of solutions to an elliptic equation involving a -Laplacian and a -Laplacian for large

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• 作者：Denis Bonheure^a ; ^b ; ^{denis.bonheure@ulb.ac.be} ; Julio D. Rossi^c ; ^{jrossi@dm.uba.ar}
• 关键词：35J20 ; 35J60 ; 35J70
• 刊名：Nonlinear Analysis: Theory, Methods & Applications
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：150
• 期：Complete
• 页码：104-113
• 全文大小：598 K
• 卷排序：150

文摘

In this paper we study the behavior as p→∞p→∞ of solutions up,qup,q to −Δpu−Δqu=0−Δpu−Δqu=0 in a bounded smooth domain ΩΩ with a Lipschitz Dirichlet boundary datum u=gu=g on ∂Ω∂Ω. We find that there is a uniform limit of a subsequence of solutions, that is, there is pj→∞pj→∞ such that upj,q→u∞upj,q→u∞ uniformly in Ω¯ and we prove that this limit u∞u∞ is a solution to a variational problem, that, when the Lipschitz constant of the boundary datum is less than or equal to one, is given by the minimization of the LqLq-norm of the gradient with a pointwise constraint on the gradient. In addition we show that the limit is a viscosity solution to a limit PDE problem that involves the qq-Laplacian and the ∞∞-Laplacian.