On the effect of critical points of distance function in superlinear elliptic problems

Abstract

We study some perturbed semilinear problems with Dirichlet or Neumann boundary conditions, $$ \begin{cases} -{\varepsilon}^2\Delta u+u=u^p & \mbox{ in $\Omega$}\cr u>0 & \mbox{ in $\Omega$}\cr u=0\ \ \mbox{ or }\ \ {{\partial u}\over{\partial\nu}}=0 & \mbox{ in $\partial\Omega,$} \end{cases} $$ 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 any ``suitable" critical point $x_0$ of the distance function generates a family of single interior spike solutions, whose local maximum point tends to $x_0$ as ${\varepsilon}$ tends to zero.