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February, 2018 Blow-up Solution for $4$-dimensional Generalized Emden-Fowler Equation with Exponential Nonlinearity
Taieb Ouni
Taiwanese J. Math. 22(1): 125-155 (February, 2018). DOI: 10.11650/tjm/170804

Abstract

Using some nonlinear domain decomposition method, we prove the existence of singular limits for solution of generalized Emden-Fowler equation with exponential nonlinearity in fourth-dimensional given by \[ \begin{cases} \Delta (a(x) \Delta u) - V(x) \operatorname{div}(a(x) \nabla u) = \rho^{4} a(x) e^{u} & \textrm{in $\Omega \subset \mathbb{R}^{4}$}, \\ u = \Delta u = 0 & \textrm{on $\partial \Omega$}. \end{cases} \] The leading part $\Delta$ is, usually, called Laplacian operator. The potential $V(x)$ belongs to $L^{\infty}_{\operatorname{loc}}(\mathbb{R}^{4})$ it is smooth and bounded and $a = a(x)$ is a given smooth function over $\overline{\Omega}$, called the Schrödinger wave function. Namely, we are still looking for solutions which concentrate at the points $x^j \in \Omega$, $j = 1,\ldots,m$ as the parameter $\rho$ tends to $0$. We find sufficient conditions under which, as $\rho$ tend to $0$, there exists an explicit class of solutions which admit a concentration behavior with a prescribed bubble profile around some given $m$-points in $\Omega$, for any given integer $m$. These are the so-called singular limits. The candidate $m$-points of concentration must be nondegenerate (in essential way) critical points of a suitable finite dimensional functional explicitly and the higher order Green's function with respect to the imposed boundary conditions.

Citation

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Taieb Ouni. "Blow-up Solution for $4$-dimensional Generalized Emden-Fowler Equation with Exponential Nonlinearity." Taiwanese J. Math. 22 (1) 125 - 155, February, 2018. https://doi.org/10.11650/tjm/170804

Information

Received: 22 March 2016; Revised: 7 August 2017; Accepted: 14 August 2017; Published: February, 2018
First available in Project Euclid: 4 October 2017

zbMATH: 06965363
MathSciNet: MR3749358
Digital Object Identifier: 10.11650/tjm/170804

Subjects:
Primary: 35J40 , 35J60 , 35J75 , 58J08

Keywords: Emden-Fowler equation , Green's function , nonlinear domain decomposition method , singular limits

Rights: Copyright © 2018 The Mathematical Society of the Republic of China

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Vol.22 • No. 1 • February, 2018
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