Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory

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.