Electron. J. Diff. Equ., Vol. 2011 (2011), No. 81, pp. 1-13.

Existence of continuous positive solutions for some nonlinear polyharmonic systems outside the unit ball

Sameh Turki

Abstract:
We study the existence of continuous positive solutions of the m-polyharmonic nonlinear elliptic system
$$\displaylines{ (-\Delta)^{m}u+\lambda p(x)g(v)=0,\cr (-\Delta )^{m}v+\mu q(x)f(u)=0 }$$
in the complement of the unit closed ball in $\mathbb{R}^{n}$ (n>2m and $m\geq 1$). Here the constants $\lambda,\mu$ are nonnegative, the functions f,g are nonnegative, continuous and monotone. We prove two existence results for the above system subject to some boundary conditions, where the nonnegative functions p,q satisfy some appropriate conditions related to a Kato class of functions.

Submitted May 23, 2011. Published June 21, 2011.
Math Subject Classifications: 34B27, 35J40.
Key Words: Polyharmonic elliptic system; Positive solutions; Green function; polyharmonic Kato class.

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Sameh Turki
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 2092 Tunis, Tunisia
email: sameh.turki@ipein.rnu.tn

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