Deformation retracts to the fat diagonal and applications to the existence of peak solutions of nonlinear elliptic equations

E. Norman Dancer, Jonathan Hillman and Angela Pistoia

Vol. 256 (2012), No. 1, 67–78

DOI: 10.2140/pjm.2012.256.67

Abstract

We consider the equation −𝜀²Δu = u^p −u^q in a bounded, smooth domain Ω ⊂ ℝ^N with homogeneous Dirichlet boundary conditions when either

We prove the existence of multiple positive solutions in the case of small diffusion provided the domain Ω is not contractible.

Keywords

multiple peaks, finite-dimensional reduction, fat diagonal

Mathematical Subject Classification 2010

Primary: 35B40, 35J20, 35J55

Milestones

Received: 15 April 2011

Revised: 22 June 2011

Accepted: 14 September 2011

Published: 6 May 2012