Blow-up solutions of some nonlinear elliptic problems. (English) Zbl 0894.35038

Brézis, Haïm; Vázquez, Juan Luis

Rev. Mat. Univ. Complutense Madr. 10, No. 2, 443-469 (1997).

The authors study properties of extremal solutions of the semilinear equation \(-\Delta u=\lambda f(u)\), posed in a bounded domain \(\Omega\subset\mathbb{R}^n\), with Dirichlet data \(u=0\) on the (smooth) boundary \(\partial\Omega\). The nonlinearity \(f\) is assumed to be continuous, positive, increasing, and convex on \(u\geq 0\) with \(f(0)>0\) and \(\lim_{u\to\infty}f(u)/u= \infty\). Under this conditions the existence of an extremal value \(\lambda^*\) is known, such that the Dirichlet problem has at least one positive classical solution if \(0<\lambda<\lambda^*\) and no solution if \(\lambda>\lambda^*\). Solutions at the value \(\lambda^*\) are referred to as extremal solutions. The authors characterize the singular (unbounded) extremal solutions and the extremal value \(\lambda^*\) by means of a version of Hardy’s inequality. To discuss typical examples (\(f(u)= e^u\), \(f(u)= (1+u)^p\), \(p>1\)), an improved version of the classical Hardy inequality is derived. The authors conclude with some open problems connected with the results of this paper.

Reviewer: Ralf Beyerstedt (Triest)

MSC:

35J65

Nonlinear boundary value problems for linear elliptic equations

Keywords:

extremal solutions; Hardy’s inequality