Uniqueness for the Semilinear Elliptic Problems

Abstract

In this paper, we study the positive solutions of the semilinear elliptic equation $$\begin{cases} Lu+g(x,u)u=0 \ \ &{\rm in}& \Omega, \\ Bu=0 \ \ &{\rm on}& ∂Ω, \end{cases}$$where $\Omega ⊂\mathbb{R}^N$ is a bounded smooth domain, $L$ is an elliptic operator, $B$ is a general boundary operator and $g(·, ·)$ is a continuous function. This is a general problem proposed by Amann [Arch. Rational Mech. Anal. 44 (1972)], Cac [J. London Math. Soc. 25 (1982)] and Hess [Math. Z. 154 (1977)]. We obtain various uniqueness results when the nonlinearity function $g$ satisfies some additional conditions.