Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory. (English) Zbl 0964.35047

Grossi, Massimo; Pistoia, Angela; Wei, Juncheng

Calc. Var. Partial Differ. Equ. 11, No. 2, 143-175 (2000).

The paper deals with the existence of positive multipeak solutions of the semilinear Neumann problem \[ -\varepsilon^2 \Delta u+u= u^p\quad \text{in}\;\Omega,\qquad \partial u/\partial\nu=0\quad \text{on}\;\partial\Omega, \] where \(\Omega\subset\mathbb R^N\) is a bounded and smooth domain, \(N\geq 2,\) \(\varepsilon >0,\) \(1<p<(N+2)/(N-2)\) if \(N\geq 3\) and \(p>1\) if \(N=2,\) and \(\nu\) is the unit outward normal to \(\partial\Omega.\)

Reviewer: Lubomira Softova (Bari)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35B38	Critical points of functionals in context of PDEs (e.g., energy functionals)
35J61	Semilinear elliptic equations

Keywords:

semilinear elliptic equation; Neumann problem; multipeak solutions