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IMA Journal of Applied Mathematics Advance Access originally published online on January 13, 2007
IMA Journal of Applied Mathematics 2007 72(2):113-139; doi:10.1093/imamat/hxl032
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

On the shape of least-energy solutions for a class of quasilinear elliptic Neumann problems

Yi Li1,** and Chunshan Zhao2,***

1 Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA, 2 Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA and Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460, USA

** Email: yli{at}math.uiowa.edu

*** Email: chuzhao{at}math.uiowa.edu


  Abstract

We study the shape of least-energy solutions to the quasilinear elliptic equation {varepsilon}m{Delta}mu – um–1 + f(u) = 0 with homogeneous Neumann boundary condition as {varepsilon} -> 0+ in a smooth bounded domain {Omega} sub RN. Firstly, we give a sharp upper bound for the energy of the least-energy solutions as {varepsilon} -> 0+, which plays an important role to locate the global maximum. Secondly, based on this sharp upper bound for the least energy, we show that the least-energy solutions concentrate on a point P{varepsilon} and dist(P{varepsilon}, {partial}{Omega})/{varepsilon} goes to zero as {varepsilon} -> 0+. We also give an approximation result and find that as {varepsilon} -> 0+ the least-energy solutions go to zero exponentially except a small neighbourhood with diameter O({varepsilon}) of P{varepsilon} where they concentrate.

Keywords: quasi-linear Neumann problem; m-Laplacian operator; least-energy solution.

Received on 16 June 2004. accepted on 18 October 2006.


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