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May  2010, 9(3): 761-778. doi: 10.3934/cpaa.2010.9.761

Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity

Liping Wang 1,

1. 

Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241, China

Received  May 2009 Revised  October 2009 Published  January 2010

We consider the sub- or supercritical Neumann elliptic problem $-\Delta u + \mu u = u^{\frac{N + 2}{N - 2} + \varepsilon}, u > 0 $ in $\Omega; \frac{\partial u}{\partial n} = 0 $ on $\partial \Omega, \Omega$ being a smooth bounded domain in $R^N, N \ge 4, \mu > 0 $ and $\varepsilon \ne 0$. Let $H(x)$ denote the mean curvature at $x$. We show that for slightly sub- or supercritical problem, if $\varepsilon \min_{x \in \partial\Omega} H(x) > 0$ then there always exist arbitrarily many solutions which blow up at the least curved part of the boundary as $\varepsilon$ goes to zero.
Keywords: sub- or supercritical nonlinearity, Arbitrarily many solutions, mean curvature..
Mathematics Subject Classification: Primary 35B35, 92C15; Secondary 35B40, 92D2.
Citation: Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure and Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761
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