# American Institute of Mathematical Sciences

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

 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.
Citation: Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761
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