The extremal solution of a boundary reaction problem. (English) Zbl 1156.35039

Dávila, Juan; Dupaigne, Louis; Montenegro, Marcelo

Commun. Pure Appl. Anal. 7, No. 4, 795-817 (2008).

Summary: We consider the problem \[ \Delta u=0\text{ in }\Omega,\qquad \frac{\partial u} {\partial\nu}=\lambda f(u)\text{ on }\Gamma_1,\qquad u=0\text{ on }\Gamma_2 \] where \(\lambda >0\), \(f(u)=e^u\) or \(f(u)= (1+u)^p\), \(\Gamma_1,\Gamma_2\) is a partition of \(\partial\Omega\) and \(\Omega\subset\mathbb{R}^N\). We determine sharp conditions on the dimension \(N\) and \(p>1\) such that the extremal solution is bounded, where the extremal solution refers to the one associated to the largest \(\lambda\) for which a solution exists.

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35B35	Stability in context of PDEs
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

boundary reaction problem; nonlinear Neumann problem; extremal solution; stability; Kato’s inequality