On the effect of critical points of distance function in superlinear elliptic problems. (English) Zbl 0989.35054

Summary: We study some perturbed semilinear problems with Dirichlet or Neumann boundary conditions, \[ \begin{cases} -\varepsilon^2 \Delta u+u= u^p\quad & \text{in }\Omega\\ u>0 & \text{in }\Omega\\ u=0\text{ or }{\partial u\over \partial v}=0 & \text{in }\partial \Omega,\end{cases} \] where \(\Omega\) is a bounded, smooth domain of \(\mathbb{R}^N\), \(N\geq 2\), \(\varepsilon >0\), \(1<p< {N+2\over N-2}\) if \(N\geq 3\) 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.


35J65 Nonlinear boundary value problems for linear elliptic equations
35B25 Singular perturbations in context of PDEs
35J20 Variational methods for second-order elliptic equations