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Zbl 0989.35054
Grossi, Massimo; Pistoia, Angela
On the effect of critical points of distance function in superlinear elliptic problems. (English)
[J] Adv. Differ. Equ. 5, No.10-12, 1397-1420 (2000). ISSN 1079-9389

Summary: We study some perturbed semilinear problems with Dirichlet or Neumann boundary conditions, $$\cases -\varepsilon^2 \Delta u+u= u^p\quad & \text{in }\Omega\\ u>0 & \text{in }\Omega\\ u=0\text{ or }{\partial u\over \partial v}=0 & \text{in }\partial \Omega,\endcases$$ where $\Omega$ is a bounded, smooth domain of $\bbfR^N$, $N\ge 2$, $\varepsilon >0$, $1<p< {N+2\over N-2}$ if $N\ge 3$ 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.

MSC 2000:: *35J65 (Nonlinear) BVP for (non)linear elliptic equations
35B25 Singular perturbations (PDE)
35J20 Second order elliptic equations, variational methods

Keywords: family of single interior spike solutions

Cited in: Zbl 1216.35042

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Zentralblatt MATH Berlin [Germany]