On the existence of positive continuous solutions for some polyharmonic elliptic systems on the half space

• Autorzy: Zine El Abidine Z.
• Języki publikacji: EN

Abstrakty

EN

We study the existence of positive continuous solutions of the nonlinear polyharmonic system (-Δ)mu + λqg(v) = 0, (-Δ)mv + μpf(u) = 0 in the half space [formula] where m ≥1 and n>2m.The nonlinear term is required to satisfy some conditions related to the Kato class [formula]. Our arguments are based on potential theory tools associated to (-Δ)m and properties of functions belonging to [formula].

Słowa kluczowe

EN

polyharmonic elliptic system; Green function; Kato class; positive continuous solution; Schauder fixed point theorem

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2012
• Tom: Vol. 32, no. 1
• Strony: 91--113
• Opis fizyczny: Bibliogr. 18 poz.

Twórcy

autor

Zine El Abidine Z.

• Département de Mathématiques Faculté des Sciences de Tunis Campus Universitaire, 2092 Tunis, Tunisia,

Bibliografia

• [1] I. Bachar, H. Mâagli, Estimates on the Green’s function and existence of positive solutions of nonlinear singular elliptic equation in the half space, Positivity 9 (2005), 153–192.
• [2] I. Bachar, H. Mâagli, L. Mâatoug, Positive solutions of nonlinear elliptic equations in a half space in R2, Electron. J. Differential Equations 41 (2002), 1–24.
• [3] I. Bachar, H. Mâagli, S. Masmoudi, M. Zribi, Estimates for the Green function and singular solutions for polyharmonic nonlinear equation, Abstr. Appl. Anal. 12 (2003), 715–741.
• [4] I. Bachar, H. Mâagli, M. Zribi, Estimates on the Green function and existence of positive solutions for some polyharmonic nonlinear equations in the half space, Manuscripta Math. 113 (2004), 269–291.
• [5] I. Bachar, M. Zribi, Existence results for some polyharmonic problems in the half-space, J. Math. Anal. Appl. 322 (2006), 610–620.
• [6] T. Boggio, Sulle funzioni di Green d’ordine m, Rend. Circ. Mat. Palermo 20 (1905), 97–135.
• [7] F.C. Cirstea, V.D. Radulescu, Entire solutions blowing up at infinity for semilinear elliptic systems, J. Math. Pures Appl. 81 (2002), 827–846.
• [8] F. David, Radial solutions of an elliptic system, Houston J. Math. 15 (1989), 425–458.
• [9] A. Ghanmi, H. Maagli, V. Radulescu, N. Zeddini, Large and bounded solutions for a class of nonlinear Schrödinger stationary systems, Anal. Appl. 7 (2009), 391–404.
• [10] A. Ghanmi, H. Mâagli, S. Turki, N. Zeddini, Existence of positive bounded solutions for some nonlinear elliptic systems, J. Math. Anal. Appl. 352 (2009), 440–448.
• [11] M. Ghergu, V.D. Radulescu, On a class of singular Gierer-Meinhart systems arising in morphogenesis, C. R. Acad. Sci. Paris. Ser. I 344 (2007), 163–168.
• [12] M. Ghergu, V. Radulescu, A singular Gierer-Meinhardt system with different source terms, Proc. Roy. Soc. Edinburgh Sect. A 138A (2008), 1215–1234.
• [13] M. Ghergu, V. Radulescu, Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Ser. Math. Appl. vol. 37, Oxford University Press, 2008.
• [14] S. Gontara, Z. Zine El Abidine, Existence of positive bounded solutions for some nonlinear polyharmonic elliptic systems, Electron. J. Differential Equations 113 (2010), 1–18.
• [15] H.C. Grunau, G. Sweers, Positivity for equations involving polyharmonic oprators with Dirichlet boundary conditions, Math. Ann. 307 (1997), 589–626.
• [16] A.V. Lair, A.W. Wood, Existence of entire large positive solutions of semilinear elliptic systems, J. Differential Equations 164 (2000) 2, 380–394.
• [17] S. Port, C. Stone, Brownian Motion and Classical Potential Theory, Probab. Math. Statist. Academic Press, New York, 1978.
• [18] D. Ye, F. Zhou, Existence and nonexistence of entire large solutions for some semilinear elliptic equations, J. Partial Differential Equations 21 (2008), 253–262.