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Existence and asymptotic behavior of positive continuous solutions for a nonlinear elliptic system in the half space

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This paper deals with the existence and the asymptotic behavior of positive continuous solutions of the nonlinear elliptic system [formula] in the half space [formula] where α, β ≥ 1and r, s .≥ 0. The functions p and q are required to satisfy some appropriate conditions related to the Kato class [formula]. Our approach is based on potential theory tools and the use of Schauder's fixed point theorem.
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783--795
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Bibliogr. 18 poz.
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Bibliografia
  • [1] D.H. Armitage, S.J. Gardiner, Classical potential theory, Springer, 2001.
  • [2] I. Bachar, H. Maagli, Estimates on the Green’s function and existence of positive solutions of nonlinear singular elliptic equations in the half space, Positivity 9 (2005), 153–192.
  • [3] I. Bachar, H. Maagli, L. Maatoug, Positive solutions of nonlinear elliptic equations in a half space in R2, Electron. J. Differential Equations 41 (2002), 1–24.
  • [4] I. Bachar, H. Mâagli, M. Zribi, Existence of positive solutions to nonlinear elliptic problem in the half space, Electron. J. Differential Equations 44 (2005), 1–16.
  • [5] K.L. Chung, Z. Zhao, From Brownian motion to Schrödinger’s equation, Springer Verlag, 1995.
  • [6] J. García-Melián, J.D. Rossi, Boundary blow-up solutions to elliptic systems of competitive type, J. Differential Equations 206 (2004), 156–181.
  • [7] J. García-Melián, A remark on uniqueness of large solutions for elliptic systems of competitive type, J. Math. Anal. Appl. 331 (2007), 608–616.
  • [8] M. Ghergu, V. Radulescu, On a class of singular Gierer-Meinhardt systems arising in morphogenesis, C. R. Math. Acad. Sci. Paris 344 (2007), 163–168.
  • [9] M. Ghergu, V. Rˇadulescu, A singular Gierer-Meinhardt system with different source terms, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 1215–1234.
  • [10] M. Ghergu, V. Radulescu, Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics, Springer Monographs in Mathematics, Springer Verlag, Heidelberg, 2011.
  • [11] A. Kristály, V. Radulescu, C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136, Cambridge University Press, Cambridge, 2010.
  • [12] H. Maagli, Perturbation semi-linéaire des résolvantes et des semi-groupes, Potential Anal. 3 (1994) 61–87.
  • [13] H. Maagli, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem, Nonlinear Anal. 74 (2011) 9, 2941–2947.
  • [14] V. Radulescu, D. Repovs, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal. 75 (2012), 1524–1530.
  • [15] I. Skrypnik, The Harnack inequality for a nonlinear elliptic equation with coefficients from the Kato class. (Russian) Ukr. Math. Visn. 2 (2005), No. 2, 219–235, 295; translation in Ukr. Math. Bull. 2 (2005) 2, 223–238.
  • [16] C. Yarur, Existence of continuous and singular ground states for semilinear ellptic systems, Electron. J. Differential Equations 1998 (1998) 1, 1–27.
  • [17] Z. Zhang, Existence of entire positive solutions for a class of semilinear elliptic systems, Electron. J. Differential Equations 2010 (2010) 16, 1–5.
  • [18] Z. Zhao, On the existence of positive solutions of nonlinear elliptic equations. A probabilistic potential theory approach, Duke Math. J. 69 (1993), 247–258.
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Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGHS-0007-0013
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