Limiting profile of the blow-up solutions for the fourth-order nonlinear Emden-Fowler equation with a singular source

Abstract

We consider the anisotropic Emden-Fowler equation with exponential nonlinearity in fourth-dimensional with a singular source, given by \begin{document}$ \begin{equation*} \left\{\begin{array}{rclll}\Delta(a(x)\Delta u)-V(x)\,div(a(x)\nabla\,u) & = & \rho^{4} a(x)\,e^{u} -32 \pi^2\;\alpha\,a(x)\,\delta_{p}&\mbox{in} \; \; \Omega\subset \mathbb R^{4}\\\\u = \Delta u & = &0 &\mbox{on} \; \; \partial\Omega. \end{array} \right. \end{equation*} $\end{document} The leading part $ \Delta $ is, usually, called Laplacian operator. $ V = V(|x|) $ is a smooth bounded radial potential and $ a = a(|x|) $ is a given smooth radial function over $ \bar{\Omega} $, called the Schrödinger wave function. Namely, we extend the result of [8] by allowing the presence in the equation a singular source given by Dirac measure with pole at point $ p $ and a generalization in higher dimensional, of the interest problem treated in [11]. We construct a family of solutions $ u_{\rho} $ that concentrate at $ p $ as $ \rho $ tends to $ 0 $.