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

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  • 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.
    Mathematics Subject Classification: Primary 35B35, 92C15; Secondary 35B40, 92D25.


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