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and applications to Schrödinger operators
Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical
nonlinearity
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.