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ISSN : 1226-0657

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Vol.17 No.3

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SOLVABILITY FOR SOME DIRICHLET PROBLEM WITH P-LAPACIAN

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2010, v.17 no.3, pp.257-268
Kim, Yong-In
Kim,, Y. (2010). SOLVABILITY FOR SOME DIRICHLET PROBLEM WITH P-LAPACIAN. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 17(3), 257-268.

Abstract

We investigate the existence of the following Dirichlet boundary value problem <TEX>$({\mid}u'\mid^{p-2}u')'\;+\;(p\;-\;1)[\alpha{\mid}u^+\mid^{p-2}u^+\;-\;\beta{\mid}u^-\mid^{p-2}u^-]$</TEX> = (p - 1)h(t), u(0) = u(T) = 0, where p > 1, <TEX>$\alpha$</TEX> > 0, <TEX>$\beta$</TEX> > 0 and <TEX>${\alpha}^{-\frac{1}{p}}\;+\;{\beta}^{-\frac{1}{p}}\;=\;2$</TEX>, <TEX>$T\;=\;{\pi}_p/{\alpha}^{\frac{1}{p}}$</TEX>, <TEX>${\pi}_p\;=\; \frac{2{\pi}}{p\;sin(\pi/p)}$</TEX> and <TEX>$h\;{\in}\;L^{\infty}$</TEX>(0,T). The results of this paper generalize some early results obtained in [8] and [9]. Moreover, the method used in this paper is elementary and new.

keywords
Dirichlet problem, p-Laplacian, Fredholm alternative

Reference

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Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics

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