Abstract.
We study a perturbed semilinear problem with Neumann boundary condition
\[ \cases{ -\varepsilon^2\Delta u+u=u^p & {\rm in} \Omega \cr &\cr u>0 & {\rm in} \Omega\cr &\cr {{\partial u}\over{\partial\nu}}=0& {\rm in} \partial\Omega,\cr} \]
where \(\Omega\) is a bounded smooth domain of \({mathbb{R}}^N\), \(N\ge2\), \(\varepsilon>0\), \(1 < p < {{N+2}\over{N-2}}\) if \(N\ge3\) or \(p>1\) if \(N=2\) and \(\nu\) is the unit outward normal at the boundary of \(\Omega\). We show that for any fixed positive integer K any “suitable” critical point \((x_0^1,\dots,x_0^K)\) of the function
\begin{eqnarray*} \lefteqn{\varphi_K(x^1,\dots,x^K)} &=& \min\left\{{\rm dist}(x^i,{\partial\Omega}),{|x^j-x^l|\over2} \mid i,j,l=1.\dots,K, j\ne l\right\} \end{eqnarray*}
generates a family of multiple interior spike solutions, whose local maximum points \(x_\varepsilon^1,\dots,x_\varepsilon^K\) tend to \(x_0^1,\dots,x_0^K\) as \(\varepsilon\) tends to zero.
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Received March 7, 1999 / Accepted October 1, 1999 / Published online April 6, 2000
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Grossi, M., Pistoia, A. & Wei, J. Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory. Calc Var 11, 143–175 (2000). https://doi.org/10.1007/PL00009907
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DOI: https://doi.org/10.1007/PL00009907