Electron. J. Diff. Equ., Vol. 2015 (2015), No. 289, pp. 1-19.

Singular limiting solutions to 4-dimensional elliptic problems involving exponentially dominated nonlinearity and nonlinear terms

Sami Baraket, Imen Bazarbacha, Maryem Trabelsi

Abstract:
Let $\Omega \in \mathbb{R}^4$ be a bounded open regular set, $x_1, x_2, \dots, x_m \in \Omega$, $\lambda, \rho >0$ and $Q_\lambda$ be a non linear operator (which will be defined later). We prove that the problem
$$ \Delta^2u +Q_\lambda(u)= \rho^4 e^u $$
has a positive weak solution in $\Omega$ with $u=\Delta u=0$ on $\partial \Omega$, which is singular at each $x_i$ as the parameters $\lambda$ and $\rho$ tends to 0.

Submitted August 15, 2015. Published November 20, 2015.
Math Subject Classifications: 35J60, 53C21, 58J05.
Key Words: Biharmonic operator; nonlinear operator; singular limits; Green's function; nonlinear domain decomposition method.

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Sami Baraket
Department of Mathematics, College of Science
King Saud University, P.O. Box 2455
Riyadh 11451, Saudi Arabia
email: sbaraket@ksu.edu.sa
Imen Bazarbacha
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 2092 Tunis
University Tunis El Manar, Tunisia
email: imen.bazarbacha@gmail.com
  Maryem Trabelsi
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 2092 Tunis
University Tunis El Manar, Tunisia
email: trabelsi.maryem@gmail.com

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