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- W3166160889 abstract "<p style='text-indent:20px;'>In this paper, we investigate a weighted Dirichlet eigenvalue problem for a class of degenerate operators related to the <inline-formula><tex-math id=M2>begin{document}$ h $end{document}</tex-math></inline-formula> degree homogeneous <inline-formula><tex-math id=M3>begin{document}$ p $end{document}</tex-math></inline-formula>-Laplacian</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=FE1> begin{document}$ begin{equation} nonumber left { begin{array}{ll} {|Du|^{h-1}}Delta_p^N u+ lambda a(x)|u|^{h-1}u = 0, quadquad rm{in}quad Omega, u = 0, quadquad quad quad rm{on} quadpartialOmega. end{array}right. end{equation} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Here <inline-formula><tex-math id=M4>begin{document}$ a(x) $end{document}</tex-math></inline-formula> is a positive continuous bounded function in the closure of <inline-formula><tex-math id=M5>begin{document}$ Omegasubset mathbb{R}^n(ngeq 2), $end{document}</tex-math></inline-formula> <inline-formula><tex-math id=M6>begin{document}$ h>1, $end{document}</tex-math></inline-formula> <inline-formula><tex-math id=M7>begin{document}$ 2< p<infty, $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=M8>begin{document}$ Delta_p^N u = frac{1}{p}|Du|^{2-p} {rm div}left(|Du|^{p-2}Duright) $end{document}</tex-math></inline-formula> is the normalized version of the <inline-formula><tex-math id=M9>begin{document}$ p $end{document}</tex-math></inline-formula>-Laplacian arising from a stochastic game named Tug-of-War with noise. We prove the existence of the principal eigenvalue <inline-formula><tex-math id=M10>begin{document}$ lambda_Omega $end{document}</tex-math></inline-formula>, which is positive and has a corresponding positive eigenfunction for <inline-formula><tex-math id=M11>begin{document}$ p>n $end{document}</tex-math></inline-formula>. The method is based on the maximum principle and approach analysis to the weighted eigenvalue problem. When a parameter <inline-formula><tex-math id=M12>begin{document}$ lambda<lambda_Omega $end{document}</tex-math></inline-formula>, we establish some existence and uniqueness results related to this problem. During this procedure, we also prove some regularity estimates including Hölder continuity and Harnack inequality.</p>" @default.
- W3166160889 created "2021-06-22" @default.
- W3166160889 creator A5085025467 @default.
- W3166160889 date "2022-01-01" @default.
- W3166160889 modified "2023-09-25" @default.
- W3166160889 title "The eigenvalue problem for a class of degenerate operators related to the normalized $ p $-Laplacian" @default.
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- W3166160889 doi "https://doi.org/10.3934/dcdsb.2021155" @default.
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