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Singular limits in higher order Liouville-type equations

  • Fabrizio MorlandoEmail author
Article
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Abstract

In this paper we consider the following higher order boundary value problem
$$\left\{ \begin{array}{l@{\quad}l} (-\Delta)^{m} u=\rho^{2m} V(x) e^{u}& \mbox{in} \ \Omega\\ B_{j}u=0, |j|\leq m-1& \mbox{on} \ \partial\Omega,\ \end{array} \right.$$
where \({\Omega}\) is a smooth bounded domain in \({\mathbb{R}^{2m}}\), \({m\in\mathbb{N}}\), \({V(x)\neq0}\) is a smooth function positive somewhere in \({\Omega}\) and \({\rho}\) is a positive small parameter. Here, the operator B j stands for either Navier or Dirichlet boundary conditions. We find sufficient conditions under which, as \({\rho}\) approaches 0, there exists an explicit class of solutions which admit a concentration behavior with a prescribed bubble profile around some given k-points in \({\Omega}\), for any given integer k. These are the so-called singular limits.

Keywords

Higher order nonlinear elliptic equation Concentrating solution Lyapunov–Schmidt reduction 

Mathematics Subject Classification

Primary 35J40 35B40 35B33 

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© Springer Basel 2015

Authors and Affiliations

  1. 1.Università degli Studi Roma TreRomeItaly

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