Uniqueness of weak extremal solutions of nonlinear elliptic problems. (English) Zbl 0884.35037

Martel, Yvan

Houston J. Math. 23, No. 1, 161-168 (1997).

This work concerns the uniqueness of positive weak solutions of the following elliptic problem \[ -\triangle u=\lambda g(u)\text{ in }\Omega,\qquad u=0\text{ on }\partial\Omega, \] where \(\Omega\subset\mathbb{R}^N\) is a smooth bounded domain, \(g:[0,\infty)\to[0,\infty)\) is a \(C^1\) convex positive nondecreasing function with \(g(0)>0\), \(g\not\equiv g(0)\) and \(\lambda>0\) is a critical value. It is used the notion of weak solution introduced by H. Brezis, T. Cazenave, Y. Martel, and A. Ramiandrisoa [Adv. Differ. Equ. 1, No. 1, 73-90 (1996; Zbl 0855.35063)], where related information can be found. A similar uniqueness result is proved for the problem \[ -\triangle u= g(u)+f(x)\text{ in }\Omega, \qquad u=0\text{ on }\partial\Omega, \] with \(f\) belonging to a class of nonnegative functions contained in \(C^{0,\alpha}(\Omega)\), \(0<\alpha<1\).

Reviewer: L.A.Fernandez (Santander)

MSC:

35J65

Nonlinear boundary value problems for linear elliptic equations

Keywords:

uniqueness of positive weak solutions

Citations:

Zbl 0855.35063